Projective characters of metacyclic p-groups

Abstract

Let ω be a 2-cocycle of a metacyclic p-group G representing a non-trivial element of the Schur multiplier $M(G).$ Then the number of ω-regular conjugacy classes of G, the subgroup consisting of the ω-regular elements in the center of G, the degree of each irreducible ω-character of G and a representation group H of G with M(H) trivial are all determined. Finally, for ω constructed from H, the projective character table of G corresponding to ω is found in the case that G is of positive type.

Communicated by Mark Lewis

Keywords:

• Metacyclic p-groups
• projective characters

2020 Mathematics Subject Classification:

• Primary: 20C25

1 Introduction

All groups considered in this paper will be finite and such a group G is metacyclic if there exists a cyclic normal subgroup N of G such that G/N is cyclic. Equivalently G is metacyclic if and only if there exists a short exact sequence $$〈1〉\to C_m\to G\to C_s\to 〈1〉$$for some $m,s\in \mathbb{N},$ where C_n denotes a cyclic group of order n (under the operation of multiplication).

Let G be a group and let ω be a 2-cocycle of G over the field of complex numbers $\mathbb{C},$ so that $\omega (1,x)=1=\omega (x,1)$ and $\omega (x,yz)\omega (y,z)=\omega (x,y)\omega (xy,z)$ for all x, y, $z\in G.$

The set of all such 2-cocycles of G form a group $Z^2(G,{\mathbb{C}}^{*})$ under multiplication. Two 2-cocycles ω and ψ are cohomologous if there exists a function $\delta :G\to {\mathbb{C}}^{*}$ such that $\psi =t(\delta )\omega ,$ where $$t(\delta )\omega (x,y)=\frac{\delta (x)\delta (y)}{\delta (xy)}\omega (x,y)$$for all $x,y\in G.$ This defines an equivalence relation on $Z^2(G,{\mathbb{C}}^{*})$ and the cohomology classes $[\omega ]$ form a finite abelian group, called the Schur multiplier $M(G).$

Definition 1.1.

Let ω be a 2-cocycle of a group G. Then $x\in G$ is ω-regular if $\omega (x,g)=\omega (g,x)$ for all $g\in C_G(x).$

Let $G_{\omega }$ be the set of ω-regular elements of G. It is easy to check that if $\psi \in [\omega ],$ then $G_{\psi }=G_{\omega }.$ Moreover, it is well-known that any conjugate of an ω-regular element is also ω-regular (see [7, Lemma 2.6.1]), so that we may refer to the ω-regular conjugacy classes of G, meaning the conjugacy classes of G that consist of ω-regular elements. However, if $x,y\in G_{\omega }$ then it does not follow in general that $xy\in G_{\omega }.$

We next define $Z_{\omega }(G)=G_{\omega }\cap Z(G),$ where Z(G) denotes the center of G. This set was studied in [12, Section 3] and in particular $Z_{\omega }(G)$ is a subgroup of G. A further refinement of this concept is the epicenter $Z^{*}(G)$ of G, which is defined by $$Z^{*}(G)={\cap }_{\left[\omega \right]\in M(G)}{Z}_{\omega }(G).$$

The group G is said to be unicentral if $Z^{*}(G)=Z(G),$ and to give a flavor of the results to come concerning metacyclic groups, we have [3, Proposition IV.4.21]:

Lemma 1.2.

Let G be a metacyclic group. Then M(G) is trivial if and only if G is unicentral.

To progress further we next introduce projective representations of G, using the general linear group $GL(n,\mathbb{C})$ of invertible n × n matrices over $\mathbb{C}.$

Definition 1.3.

Let ω be a 2-cocycle of a group G. Then $P:G\to GL(n,\mathbb{C})$ is an ω-representation of G if $$P(1)=I_n \text{and} P(x)P(y)=\omega (x,y)P(xy)$$

for all $x,y\in G.$

An ω-representation is also called a projective representation of G with 2-cocycle ω and its trace function is its ω-character.

The ω-representation P is said to be irreducible if there does not exist an invertible matrix M such that $MP(g)M^{-1}=P_1(g)\oplus P_2(g)$ for all $g\in G,$ where P₁ and P₂ are also ω-representations of G. The set of irreducible ω-characters is denoted by $\text{Proj}(G,\omega )$ (or $\mathit{Irr}(G)$ in the case ω is trivial) and the cardinality of this set is equal to the number of ω-regular classes of G (see [8, Theorem 1.3.6]). It is also the case that $x\notin G_{\omega }$ if and only if $\xi (x)=0$ for all $\xi \in \text{Proj}(G,\omega )$ or equivalently that $x\in G_{\omega }$ if and only if there exists $\xi \in \text{Proj}(G,\omega )$ such that $\xi (x)\cancel{=}0$ (see [8, Proposition 1.6.4]).

Given a 2-cocycle ψ of G it is well-known (see [4, Lemma 4.5]) that one may choose $\omega \in [\psi ]$ such that the order of ω equals the order of $[\psi ]$ and also so that ω is a class-preserving 2-cocycle of G. The latter condition means that the ω-characters of G are constant on the conjugacy classes of G. For such a choice satisfying both conditions the normal “row” orthogonality relations hold for the irreducible ω-characters of G and the normal “column” relations hold when restricted to $G_{\omega }.$ Here row and column refer to the rows and columns of the square projective character table of G, in which any columns corresponding to non-ω-regular conjugacy classes are omitted. Generally we will always assume that we are starting with a 2-cocycle ω with the two properties given above.

One of the ways of finding representative projective character tables of a group G is to find the ordinary character table of a representation group H for G.

Definition 1.4.

Let G be a group. Then a group H is called a representation or covering group for G if there exists $A\le Z(H)\cap H\prime$ such that $A\cong M(G)$ and $H/A\cong G.$

Details of the following basic concepts may be found in [7] or [4]. A representative class-preserving 2-cocycle ω for each $[\omega ]\in M(G)$ can be constructed from a representation group H of G using each linear character $\lambda \in \mathit{Irr}(A)$ and the order of ω is also equal to that of $[\omega ]$. This is achieved using the transgression isomorphism $\text{tra}:\mathit{Irr}(A)\to M(G).$ In fact $\text{tra}(\lambda )$ produces a specific ω for a fixed choice of conjugacy-preserving transversal of A in H but the cohomology class of ω is independent of the choice made. Moreover the elements of $\text{Proj}(G,\omega )$ are linearized by $\mathit{Irr}(H|\lambda ),$ where the latter set denotes the set of $\chi \in \mathit{Irr}(H)$ such that λ is a constituent of ${\chi }_A.$ It is also the case that $Z^{*}(G)\cong Z(H)/A$ from [12, Theorem 3.5].

Given ω constructed from $\lambda \in \mathit{Irr}(A)$ as above, we set $K=\ker \lambda$ and $\overline{H}=H/K.$ Then $\overline{H}$ is called an ω-covering group for G and ω in the sense that ω may be constructed from (with the obvious notation) $\overline{\lambda }\in \mathit{Irr}(\overline{A})$ and $\text{Proj}(G,\omega )$ are linearized by $\mathit{Irr}(\overline{H}|\overline{\lambda }).$ Again it is the case that $Z_{\omega }(G)\cong Z(\overline{H})/\overline{A},$ this time $\text{tra}$ defines an isomorphism from $\mathit{Irr}(\overline{A})$ to $〈[\omega ]〉.$

In Section 2 we will consider groups G for which $G_{\omega }=Z_{\omega }(G)$ and show that the elements of $\text{Proj}(G,\omega )$ are particularly easy to describe in this special case, which in particular covers all abelian groups. Another observation in this situation is that any 2-cocycle ω of G is automatically class-preserving, since the only ω-regular conjugacy classes of G consist of central elements. In Section 3 we consider metacyclic p-groups and in particular the division of these groups into positive and negative type when $p=2.$ In Sections 4 and 5 the Schur multiplier, a representation group, the ω-regular elements and the corresponding projective character tables for all $[\omega ]\ne [1]$ are found for any metacyclic p-group of positive type. In Section 6 similar calculations are performed for a metacyclic 2-group G of negative type, this time the Schur multiplier, a representation group, the number of ω-regular conjugacy classes, $Z_{\omega }(G)$ and the degree of each element of $\text{Proj}(G,\omega )$ are all described.

2 Groups in which the ω-regular elements are all central

Let G be a group and let N be a normal subgroup of G. Then the inflation homomorphism, $\inf :M(G/N)\to M(G)$ is defined by $\inf ([\overline{\omega }])=[\omega ],$ where $\omega (x,y)=\overline{\omega }(xN,yN)$ for all $x,y\in G.$ For notational convenience we will also define $\omega =\inf (\overline{\omega })$ in this context.

Lemma 2.1.

Let ω be a 2-cocycle of a group G, let $U=Z_{\omega }(G)$ and let $x\in Z(G).$ Then

1. $[{\omega }_U]$ is trivial;

2. $[\omega ]$ is in the image of $\inf :M(G/U)\to M(G);$

3. if $\xi \in \text{Proj}(G,\omega )$, then ${\xi }_U=\xi (1)\mu$ for some $\mu \in \text{Proj}(U,{\omega }_U);$

4. $x\in U$ if and only if $\xi (x)\cancel{=}0$ for all $\xi \in \text{Proj}(G,\omega );$

5. $x\notin U$ if and only if $\xi (x)=0$ for some $\xi \in \text{Proj}(G,\omega );$

6. if $\xi \in \text{Proj}(G,\omega ),$ then there exists $[\overline{\psi }]\in M(G/U)$ with $[\omega ]=\inf ([\overline{\psi }])$ such that $\xi =\zeta \kappa$ for $\overline{\zeta }\in \mathit{Proj}(G,\overline{\psi })$ and some $\kappa \in \mathit{Proj}(G,\omega {\psi }^{-1})$, where if ${\xi }_U=\xi (1)\mu$ then ${\kappa }_U=\mu .$

Proof.

Parts (b), (c), (d), and (e) are all proved in [12, Theorems 4.4 and 3.4]. Part (a) can be proved in a variety of ways, for example $o([\omega ])$ divides $\xi (1)$ for all $\xi \in \text{Proj}(G,\omega ),$ and $\mu (1)=1$ in part (c). Part (f) is just an application of Clifford’s Theorem for projective characters in this special case (see [8, Theorem 2.2.1]). □

If ω is a 2-cocycle of a group G, then G is said to be of ω-central type if $|\text{Proj}(G,\omega )|=1.$ This means that the corresponding projective character table of G has just one single entry of $\sqrt{\left|G\right|}.$ We also note in passing that if $[\omega ]$ is trivial, then $\text{Proj}(G,\omega )=\{\delta \chi :\chi \in \mathit{Irr}(G)\}$ for $\delta \in \text{Proj}(G,\omega )$ with $\delta (1)=1.$

Proposition 2.2.

Let ω be a 2-cocycle of a group G such that $G_{\omega }=Z_{\omega }(G).$ Then, with $U=Z_{\omega }(G),$

1. $\xi {(1)}^2=|G:U|$ for all $\xi \in \text{Proj}(G,\omega );$

2. $[\omega ]=\inf ([\overline{\psi }])$ for some $[\overline{\psi }]\in M(G/U)$ and G/U is of $\overline{\psi }$-central type for all such $\overline{\psi }.$

Proof.

Using the notation of Lemma 2.1 (f), $\kappa (1)=1$ and so $\overline{\zeta }(gU)=0$ for all $g\notin U$ and hence G/U is of $\overline{\psi }$-central type. Finally $\xi {(1)}^2=\overline{\zeta }{(U)}^2=|G:U|.$ □

We may vary Proposition 2.2 by replacing ω by $\inf (\overline{\psi })$ as shown in Lemma 2.1 (f), so that $\text{Proj}(U,{\omega }_U)=\mathit{Irr}(U).$ The price for doing this is that the order of the original 2-cocycle ω now divides that of $\overline{\psi }$ and $\psi .$ This change of 2-cocycle clearly doesn’t affect part (a) of Proposition 2.2, but in general in part (f) of Lemma 2.1 the 2-cocycle ψ that arises depends on the choice of ξ and in particular $\omega {\psi }^{-1}$ is not always trivial.

Our next result demonstrates that the variation considered above can have other drawbacks.

Lemma 2.3.

Let ω be a 2-cocycle of a group G. Let $U=Z_{\omega }(G)$ and $V=Z_{{\omega }^n}(G)$ for $n\in \mathbb{N},$ and let $\underset{U}{\inf },\underset{V}{\inf }$ denote the inflation homomorphism from $M(G/U)$ and $M(G/V)$ respectively into $M(G).$ Then

1. $G_{\omega }\subseteq G_{{\omega }^n};$

2. there exists $[\overline{\psi }]\in M(G/U)$ and $[\overline{\varphi }]\in M(G/V)$ such that

3. $\underset{U}{\inf }([\overline{\psi }])=[\omega ]$ and $\underset{V}{\inf }([\overline{\varphi }])=\underset{U}{\inf }({[\overline{\psi }]}^n).$

Proof.

For (a), it is obvious that if $g\in G$ is ω-regular, then it is also ωⁿ-regular.

For (b), from Lemma 2.1(b) there exists $[\overline{\psi }]\in M(G/U)$ such that $\underset{U}{\inf }([\overline{\psi }])=[\omega ].$ Now $${\left[\omega \right]}^n=\underset{U}{\overset{n}{\inf }}([\overline{\psi }])=\underset{U}{\inf }({\left[\overline{\psi }\right]}^n).$$

Next every element of V is ωⁿ-regular and so every element of V is ψⁿ-regular. It follows from Lemma 2.1(d) that every element of V/U is ${\overline{\psi }}^n$-regular Thus from Lemma 2.1(b) there exists $[\overline{\varphi }]\in M((G/U)/(V/U))$ with $\inf ([\overline{\varphi }])={[\overline{\psi }]}^n.$ Finally, identifying the group $(G/U)/(V/U)$ with G/V yields the desired result. □

In the context of Lemma 2.3 we have shown that we may replace ω by $\underset{U}{\inf }(\overline{\psi }),$ but it may not be the case that we can also replace ωⁿ by $\underset{V}{\inf }({\overline{\psi }}^n).$

3 Metacyclic p-groups

Metacyclic groups G have presentations of the form $$G=〈a,b:a^m=1,b^s=a^t,ba{b}^{-1}=a^r〉,$$where m, r, s, and t are positive integers satisfying $r^s\equiv 1\hspace{1em}(\mathrm{mod}m)$ and $m|t(r-1).$ Conversely if the positive integers $m,s,r,t$ satisfy these two conditions, then the given presentation defines a metacyclic group of order ms. If G has the above presentation, then we may replace a by $a^k,$ where $k=t/(m,t),$ which means that we can, and will, assume that $t|m.$ Now Beyl, (see [3, pp. 194–195]), defines $t_0=m/(m,r-1)$ and $\lambda =t/t_0$, so that G has parameters m, r, s and $\lambda .$

It is convenient to deal with cyclic groups separately at this juncture. For if G is any cyclic group then M(G) is trivial (see [6, Proposition 2.1.1]) and the elements of $\mathit{Irr}(G)$ are well-known. To save repetition henceforth the generators of a non-cyclic metacyclic group G will normally be denoted by a and b.

In [9, Section 1] a standard parametric presentation is given of non-cyclic metacyclic p-groups using the following notation: $$G_p(\alpha ,\beta ,ϵ,\gamma ,\pm )=〈a,b:a^{p^{\alpha }}=1,b^{p^{\beta }}=a^{p^{\alpha -ϵ}},ba{b}^{-1}=a^{p^{\alpha -\gamma }\pm 1}〉,$$with details as follows for firstly abelian and secondly non-abelian groups.

Lemma 3.1.

Let G be a non-cyclic abelian metacyclic p-group. Then there exist $\alpha ,\beta \in \mathbb{N}$ and ϵ a non-negative integer with $ϵ<\alpha$ such that for an odd prime number p, $G\cong G_p(\alpha ,\beta ,ϵ,0,+);$ whereas for $p=2,G\cong G_2(\alpha ,\beta ,ϵ,0,+)$ for $\alpha \cancel{=}1$ or $G\cong G_2(1,\beta ,0,0,-).$

We note that, using the exponent of G and the classification of abelian groups, we have for G in Lemma 3.1 and $\alpha \ge \beta +ϵ$ that $G=G_p(\alpha ,\beta ,ϵ,0,+)\cong G_p(\alpha ,\beta ,0,0,+)$ and for $\alpha <\beta +ϵ$ that $G=G_p(\alpha ,\beta ,ϵ,0,+)\cong G_p(\beta +ϵ,\alpha -ϵ,0,0,+)$ for all prime numbers p. So in particular for p odd, a non-cyclic abelian metacyclic p-group is isomorphic to exactly one group of the form $G_p(\alpha ,\beta ,0,0,+)$ for $\alpha \ge \beta \ge 1.$ The next result is now obvious from these remarks.

Lemma 3.2.

Let $\alpha ,\beta \in \mathbb{N}$ and let ϵ be a non-negative integer with $ϵ<\alpha .$ Then

1. for p odd, $G_p(\alpha ,\beta ,ϵ,0,+)$ is a non-cyclic abelian metacyclic p-group of order $p^{\alpha +\beta };$

2. for $p=2,G_2(\alpha ,\beta ,ϵ,0,+)$ is a non-cyclic abelian metacyclic 2-group of order $2^{\alpha +\beta }$ for $\alpha \cancel{=}1$ and $G_2(1,\beta ,0,0,-)$ is a non-cyclic abelian metacyclic 2-group of order $2^{1+\beta }.$

We now turn to non-abelian groups where the next result is [1, Theorem 1.2].

Theorem 3.3.

Let G be a non-abelian metacyclic p-group. Then there exist non-negative integers $\alpha ,\beta ,$ γ, and ϵ with $\alpha ,\beta >0,\gamma \le \min \{\alpha -1,\beta \}$ and $\gamma +ϵ\le \alpha$ such that for an odd prime number p, $$G\cong G_p(\alpha ,\beta ,ϵ,\gamma ,+).$$

If $p=2,$ then in addition $\alpha -\gamma >1$ and $$G\cong G_2(\alpha ,\beta ,ϵ,\gamma ,+) \text{or} G_2(\alpha ,\beta ,ϵ,\gamma ,-),$$

where in the second case $ϵ\le 1.$ Moreover, if $G\cong G_p(\alpha ,\beta ,ϵ,\gamma ,+)$ then $\gamma >0$ for all prime numbers p.

A non-cyclic metacyclic p-group G is of positive type if $G\cong G_p(\alpha ,\beta ,ϵ,\gamma ,+)$ for any prime number p and of negative type if p = 2 and $G\cong G_2(\alpha ,\beta ,ϵ,\gamma ,-),$ for some parameters satisfying the conditions in Lemma 3.1 or Theorem 3.3. It is possible in the case p = 2 for G to be both of positive and negative type for different parameters and this is discussed below.

We show that if the conditions on the parameters $\alpha ,\beta ,$ ϵ and γ in Theorem 3.3 are satisfied then $G_p(\alpha ,\beta ,ϵ,\gamma ,\pm )$ defines a metacyclic p-group. The proof utilizes the p-adic valuation $v_p(n)$ of an integer.

Lemma 3.4.

Let $\alpha ,\beta ,$ γ and ϵ be non-negative integers with $\alpha ,\beta >0,\gamma \le \min \{\alpha -1,\beta \}$ and $\gamma +ϵ\le \alpha .$ Then

1. $G_p(\alpha ,\beta ,ϵ,\gamma ,+)$ is a non-abelian metacyclic group of order $p^{\alpha +\beta }$ if $\gamma >0$ and if in addition for $p=2,\alpha -\gamma >1;$

2. $G_2(\alpha ,\beta ,ϵ,\gamma ,-)$ is a non-abelian metacyclic group of order $2^{\alpha +\beta }$ if $\alpha -\gamma >1$ and $ϵ\le 1.$

Proof.

We first note that $\alpha -ϵ\ge 0,$ so $p^{\alpha -ϵ}|p^{\alpha };$ also if $p=2,$ then $\alpha -\gamma >1$ so $2^{\alpha -\gamma }-1>1.$ Now we show that the conditions ensure that ${(p^{\alpha -\gamma }\pm 1)}^{p^{\beta }}\equiv 1\hspace{1em}(\mathrm{mod}{p}^{\alpha })$ and $p^{\alpha }|p^{\alpha -ϵ}((p^{\alpha -\gamma }+1)-1)$ in case (a) or $2^{\alpha }|2^{\alpha -ϵ}((2^{\alpha -\gamma }-1)-1)$ in case (b).

First we observe that $ϵ+\gamma \le \alpha ,$ so in case (a) $p^{\alpha -ϵ}((p^{\alpha -\gamma }+1)-1)=p^{2\alpha -(\gamma +ϵ)}$ and this is divisible by $p^{\alpha }.$ On the other hand in case (b), $ϵ\in \{0,1\}$ and $ϵ+\gamma \le \alpha ,$ so $2^{\alpha -ϵ}(2^{\alpha -\gamma }-2)=2^{2\alpha -(ϵ+\gamma )}-2^{\alpha -ϵ+1}=2^{\alpha }(2^{\alpha -(ϵ+\gamma )}-2^{1-ϵ})$ and this is divisible by $2^{\alpha }.$ Now $\beta -\gamma \ge 0$ and for $p=2,\alpha -\gamma >1,$ so $v_p({(p^{\alpha -\gamma }\pm 1)}^{p^{\beta }}-1)=\alpha -\gamma +\beta \ge \alpha$ from [1, Lemma 2.3]. □

Now any metacyclic 2-group of nilpotency class at least three is not of both positive and negative type from [1, Lemma 3.1]. We observe that, up to isomorphism, $G_2(1,1,0,0,-)\cong C_2\times C_2$ is the only group in Lemma 3.1 that is only of negative type, whereas $G_2(1,\beta ,0,0,-)\cong C_2\times C_{2^{\beta }}$ for $\beta >1$ are the only groups in that lemma that are both of positive and negative type. It also follows from [1, Corollary 2.6] that, up to isomorphism, $G_2(2,\beta ,0,0,-)$ and $G_2(2,\beta ,1,0,-)$ are the only groups of nilpotency class two in Theorem 3.3 that are of negative type. Beuerle [1, Section 1] assigns a unique set of parameters for all non-abelian metacyclic p-groups up to isomorphism, however we only need to consider 2-groups of nilpotency class one or two.

Lemma 3.5.

A non-cyclic abelian metacyclic 2-group is isomorphic to exactly one group in the following list of such groups: $G_2(\alpha ,\beta ,0,0,+)$ for $\alpha \ge \beta \ge 1$ and $(\alpha ,\beta )\ne (1,1)$ or $G_2(1,1,0,0,-).$

We next consider metacyclic 2-groups of nilpotency class two, where the results are a slight variation of [1, Theorem 1.1].

Theorem 3.6.

Let G be a metacyclic 2-group of nilpotency class two. Then G is isomorphic to exactly one group on the following list:

1. $G_2(\alpha ,\beta ,0,\gamma ,+),$ where $\alpha ,\beta ,\gamma \in \mathbb{N}$ with $\alpha \ge 2\gamma ,\beta \ge \gamma$ and $(\alpha ,\gamma )\cancel{=}(2,1).$

2. $G_2(2,\beta ,0,0,-)$ for $\beta \in \mathbb{N}.$

3. $G_2(2,1,1,0,-),$ the quaternion group of order 8.

Notice that this result implies that the groups $G_2(2,\beta ,1,0,-)$ for $\beta >1$ are of both positive and negative type.

Lemma 3.7.

The groups $G_2(2,\beta ,0,0,-)$ for $\beta \in \mathbb{N}$ and $G_2(2,1,1,0,-)$ are not of positive type.

Proof.

Let $G=G_2(2,1,0,0,-)$ or $G_2(2,1,1,0,-).$ Then $Z(G)=〈a^2〉$ and so if N is a cyclic normal subgroup of G with G/N cyclic, then $\left|N\right|=4.$

Now suppose $G=G_2(2,\beta ,0,0,-)$ with $\beta \ge 2.$ Then $G\prime =〈a^2〉$ from [1, Proposition 2.5] and $G/G\prime \cong C_2\times C_{2^{\beta }}.$ Suppose N is a cyclic normal subgroup of G with $8\left|\right|N|$ and G/N cyclic. Let $a^i{b}^j\in N$ of order 8. Then ${(a^i{b}^j)}^{4}G\prime =(a^{4i}G\prime )(b^{4j}G\prime )=b^{4j}G\prime =G\prime ,$ but ${(a^i{b}^j)}^{2}G\prime =(a^{2i}G\prime )(b^{2j}G\prime )=b^{2j}G\prime \ne G\prime$ and we conclude that $j=\beta -2.$ Now from [1, Lemma 2.4], ${(a^i{b}^{2^{\beta -2}})}^4=a^{ik},$ where $k={\sum }_{j=0}^3{3}^{2^{\beta -2}j},$ but $v_2(k)\ge 2$ from [1, Lemma 2.3]. Thus ${(a^i{b}^{2^{\beta -2}})}^4=1$ and hence no such N exists. □

In general we will consider metacyclic p-groups whose parameters just satisfy the conditions in Lemma 3.1 and Theorem 3.3. The degrees of freedom that these parameters afford are discussed in detail in [1, Section 3] for groups of nilpotency class at least three.

Finally in this section we provide a conversion between Beuerle’s notation in [1] and that of Beyl: $m=p^{\alpha },s=p^{\beta },t=p^{\alpha -ϵ}$ and $r=p^{\alpha -\gamma }\pm 1$. For groups of positive type, this leads to $t_0=p^{\alpha }/(p^{\alpha },p^{\alpha -\gamma })=p^{\gamma }$ and $\lambda =t/t_0=p^{\alpha -(\gamma +ϵ)}$. For groups of negative type we know that p = 2 and $ϵ\in \{0,1\}$. Therefore $t_0=2^{\alpha }/(2^{\alpha },2^{\alpha -\gamma }-2)=2^{\alpha -1}$ and $\lambda =2^{1-ϵ}$.

4 Multipliers and representation groups of metacyclic p-groups of positive type

To save repetition we will always assume in this section and Sections 5 and 6 that any given metacyclic p-group is non-cyclic. We also shorten the notation $G_p(\alpha ,\beta ,ϵ,\gamma ,+)$ to $G_p(\alpha ,\beta ,ϵ,\gamma ),$ since we will only be working with groups of positive type in this and the next section.

We first present a known result concerning the Schur multiplier of metacyclic groups (see [6, Theorem 2.11.3]).

Theorem 4.1.

Let G be a metacyclic group, say $$G=〈a,b:a^m=1,b^s=a^t,ba{b}^{-1}=a^r〉,$$where the positive integers m, r, s and t satisfy $$r^s\equiv 1\hspace{1em}\left(\mathrm{mod}m\right),m|t(r-1) \text{and} t|m.$$

Then $M(G)\cong C_n,$ where $$n=\frac{(r-1,m)l}{m} \text{and} l=(1+r+\cdots +r^{s-1},t).$$

Using the conversion formula at the end of Section 3 we now obtain the following result for metacyclic p-groups of positive type (the calculation for groups of negative type will be postponed until Section 6).

Corollary 4.2.

Let $G\cong G_p(\alpha ,\beta ,ϵ,\gamma ).$ Then $$M(G)\cong \{\begin{array}{cc}{C}_{p^{\beta -\gamma }},\hfill & \text{if}\mathrm{ }\alpha \ge \beta +ϵ;\hfill \\ C_{p^{\alpha -(ϵ+\gamma )}},\hfill & \text{if}\mathrm{ }\alpha <\beta +ϵ.\hfill \end{array}$$

Proof.

Applying the conversion described in Section 3 to Theorem 4.1 we obtain that $M(G)\cong C_n$ where $$n=\frac{(r-1,m)l}{m}=\frac{(p^{\alpha -\gamma },p^{\alpha })l}{p^{\alpha }}=\frac{l}{p^{\gamma }},$$with $l=({\sum }_{i=0}^{p^{\beta }-1}{(p^{\alpha -\gamma }+1)}^i,p^{\alpha -ϵ})$. Now $$v_p\left(\sum _{i=0}^{p^{\beta }-1}{(p^{\alpha -\gamma }+1)}^i\right)=\beta ,$$ from [1, Lemma 2.3] and so $$l=p^{\min \{\beta ,\alpha -ϵ\}}=\{\begin{array}{cc}{p}^{\beta },\hfill & \text{if}\mathrm{ }\alpha \ge \beta +ϵ;\hfill \\ p^{\alpha -ϵ},\hfill & \text{if}\mathrm{ }\alpha <\beta +ϵ.\hfill \end{array}$$

Beyl [2] called a group G with M(G) trivial a Schur group and we adopt this terminology. So as observed in Section 3 all cyclic groups are Schur groups. Using Corollary 4.2 we can classify all (non-cyclic) metacyclic p-groups of positive type that are Schur groups. This result confirms part of the classification of all non-cyclic metacyclic p-groups that are Schur groups given in [2, Section 5].

Corollary 4.3.

Let $G=G_p(\alpha ,\beta ,ϵ,\gamma ).$ Then G is a Schur group if $\gamma =\beta$ for $\alpha \ge \beta +ϵ,$ or $ϵ=\alpha -\gamma$ for $\alpha <\beta +ϵ.$ The non-isomorphic metacyclic p-groups of positive type that are Schur groups are $G_p(\alpha ,\beta ,0,\beta ),$ and $G_p(\alpha ,\beta ,\alpha -\gamma ,\gamma )$ for $\gamma <\beta .$

Proof.

The first part is immediate from Corollary 4.2 once we note that if $\alpha \ge \beta +ϵ,$ then $G\cong G_p(\alpha ,\beta ,0,\gamma )$ from [1, Proposition 3.6]. For the second part if $\gamma =\beta$ and $ϵ=0,$ then $\alpha \ge \beta +ϵ.$ On the other hand if $ϵ=\alpha -\gamma ,$ then $\beta +ϵ=\alpha +(\beta -\gamma )\ge \alpha$ with equality if and only if $\beta =\gamma .$ In the latter case we obtain $G_p(\alpha ,\beta ,\alpha -\beta ,\beta )\cong G_p(\alpha ,\beta ,0,\beta )$ again from [1, Proposition 3.6]. Finally the groups $G_p(\alpha ,\beta ,0,\beta ),$ and $G_p(\alpha ,\beta ,\alpha -\gamma ,\gamma )$ for $\gamma <\beta$ are all non-isomorphic from the remark before Lemma 3.2 and then also Lemma 3.5, Theorem 3.6 and [1, Propositions 3.2, 3.5 and 3.6]. □

Next we turn our attention to representation groups, for which the main result we use is [3, Theorem 2.10].

Theorem 4.4.

Every metacyclic group has a metacyclic group which is a Schur group among its representation groups.

In the context of Theorem 4.4 it should be noted that an example of a metacyclic group with two non-isomorphic representation groups that are also Schur groups is given in [3, Example IV.2.12]. We will make a definitive choice for a representation group that is also a Schur group for each metacyclic p-group of positive type with a given set of parameters.

The following result, [1, Proposition 2.5], will subsequently prove very useful.

Lemma 4.5.

Let $G=G_p(\alpha ,\beta ,ϵ,\gamma ).$ Then

1. $Z(G)=〈a^{p^{\gamma }},b^{p^{\gamma }}〉$ and $|Z(G)|=p^{\alpha +\beta -2\gamma },$

2. $G\prime =〈a^{p^{\alpha -\gamma }}〉$.

Theorem 4.6.

Let $G=G_p(\alpha ,\beta ,ϵ,\gamma )$ with $\alpha \ge \beta +ϵ.$ Define the sequence

1. $H(i)=G_p(\beta +i,\beta ,ϵ,\gamma +\beta -\alpha +i)$ for $i=0,\dots ,\alpha -\gamma$ if $\beta >\alpha -\gamma ;$

2. $H(i)=G_p(\alpha -\gamma +i,\beta ,ϵ,i)$ for $i=0,\dots ,\beta$ if $\beta \le \alpha -\gamma .$

Then H(i) is a metacyclic p-group and $|M(H(i))|=p^{\alpha -\gamma -i}$ in case (a) and $p^{\beta -i}$ in case (b). Suppose $[\omega ]\in M(H(i))$ of order $p^j.$ Then $H(i+j)$ is an ω-covering group for $H(i),$ in particular the last group in each sequence is a Schur group that is a representation group for all the groups in the sequence.

Proof.

In case (a), $\beta +i\ge \beta \ge 0.$ By definition $\gamma +\beta -\alpha +i>0,$ also $\beta -(\gamma +\beta -\alpha +i)=\alpha -\gamma -i\ge 0$ and $(\beta +i)-(\gamma +\beta -\alpha +i+ϵ)=\alpha -(\gamma +ϵ).$ Thus, H(i) is a metacyclic p-group from Lemma 3.2, $|M(H(i)|=p^{\alpha -\gamma -i}$ from Corollary 4.2 and in particular $H(\alpha -\gamma )$ is a Schur group.

Let $H(i+j)=〈x,y〉.$ Then by Lemma 4.5, $Z(H(i+j))=〈x^{p^{\gamma +\beta -\alpha +i+j}},y^{p^{\gamma +\beta -\alpha +i+j}}〉$ and $H(i+j)\prime =〈x^{p^{(\beta +i+j)-(\gamma +\beta -\alpha +i+j)}}〉=〈x^{p^{\alpha -\gamma }}〉$. Therefore, $Z(H(i+j))\cap H(i+j)\prime =〈x^{p^k}〉$, where $k=\max \{\gamma +\beta -\alpha +i+j,\alpha -\gamma \}\le \max \{\beta ,\alpha -\gamma \}=\beta$. Thus, $A_i=〈x^{p^{\beta +i}}〉\le Z(H(i+j))\cap H(i+j)\prime$. Next set $a=x〈x^{p^{\beta +i}}〉$ and $b=y〈x^{p^{\beta +i}}〉$ and observe that $H(i+j)/A_i\cong H(i).$ Finally, $\left|A_i\right|=p^{(\beta +i+j)-(\beta +i)}=p^j=o([\omega ])$ and if we let ${\text{tra}}_i:\mathit{Irr}(A_i)\to M(H(i))$ be the transgression homomorphism, then $[\omega ]\in \text{Im}({\text{tra}}_i)$ since $M(H(i))$ is cyclic.

In case (b), $\alpha -\gamma +i\ge \beta \ge 0$ and $(\alpha -\gamma +i)-(i+ϵ)=\alpha -(\gamma +ϵ).$ Thus, H(i) is a metacyclic p-group from Section 3, $|M(H(i)|=p^{\beta -i}$ from Corollary 4.2 and in particular $H(\beta )$ is a Schur group.

Using the same notation as in case (a), we have by Lemma 4.5 that $Z(H(i+j))=〈x^{p^{i+j}},y^{p^{i+j}}〉$ and $H(i+j)\prime =〈x^{p^{\alpha -\gamma }}〉$. Therefore, $Z(H(i+j))\cap H(i+j)\prime =〈x^{p^{\alpha -\gamma }}〉.$ Thus $A_i=〈x^{p^{\alpha -\gamma +i}}〉\le Z(H(i+j))\cap H(i)\prime$. Next set $a=x〈x^{p^{\alpha -\gamma +i}}〉$ and $b=y〈x^{p^{\alpha -\gamma +i}}〉$ and observe that $H(i+j)/A_i\cong H(i).$ Finally, $\left|A_i\right|=p^{(\alpha -\gamma +i+j)-(\alpha -\gamma +i)}=p^j=o([{\omega }_j])$ and the subsequent argument is the same as in case (a). □

Notice in Theorem 4.6 that $G=G_p(\alpha ,\beta ,ϵ,\gamma )\cong G_p(\alpha ,\beta ,0,\gamma )$ from [1, Proposition 3.6], so that ϵ can be taken to be 0 in the statement and proof of the result. The group G is obtained in cases (a) and (b) in Theorem 4.6 by setting $i=\alpha -\beta$ and γ, respectively and the definitive choice we make for an ω_j-covering group for G is $H(\alpha -\beta +j)$ and $H(\gamma +j)$, respectively, where $[{\omega }_j]$ has order p^j and $0\le j\le \beta -\gamma .$

Theorem 4.7.

Let $G=G_p(\alpha ,\beta ,ϵ,\gamma )$ with $\alpha <\beta +ϵ.$ Define the sequence

1. $H(i)=G_p(\alpha -ϵ+i,\beta ,i,\gamma -ϵ+i)$ for $i=0,\dots ,\alpha -\gamma$ if $\gamma >ϵ;$

2. $H(i)=G_p(\alpha -\gamma +i,\beta ,ϵ-\gamma +i,i)$ for $i=0,\dots ,\alpha -ϵ$ if $\gamma \le ϵ.$

Then H(i) is a metacyclic p-group and $|M(H(i))|=p^{\alpha -\gamma -i}$ in case (a) and $p^{\alpha -ϵ-i}$ in case (b). Suppose $[\omega ]\in M(H(i))$ of order $p^j.$ Then $H(i+j)$ is an ω-covering group for $H(i),$ in particular the last group in each sequence is a Schur group that is a representation group for all the groups in the sequence.

Proof.

In case (a), $\alpha -ϵ+i\ge \alpha -ϵ>0.$ By definition $\gamma -ϵ+i>0,$ also $\beta -(\gamma -ϵ+i)\ge \beta -(\alpha -ϵ)>0,$ $(\alpha -ϵ+i)-(\gamma -ϵ+i)=\alpha -\gamma$ and $\beta +i-(\alpha -ϵ+i)>0.$ Thus H(i) is a metacyclic p-group from Lemma 3.2, $|M(H(i)|=p^{\alpha -\gamma -i}$ from Corollary 4.2 and in particular $H(\alpha -\gamma )$ is a Schur group.

Now let $H(i+j)=〈x,y〉.$ Then by Lemma 4.5, $Z(H(i+j))=〈x^{p^{\alpha -(ϵ+i+j)}},y^{p^{\alpha -(ϵ+i+j)}}〉$ and $H(i+j)\prime =〈x^{p^{\alpha -\gamma }}〉$. Therefore, $Z(H(i+j))\cap H(i+j)\prime =〈x^{p^k}〉$, where $k=\max \{\alpha -(ϵ+i+j),\alpha -\gamma \}\le \max \{\alpha -ϵ,\alpha -\gamma \}=\alpha -ϵ$. Thus, $A_i=〈x^{p^{\alpha -ϵ+i}}〉\le Z(H(i+j))\cap H(i+j)\prime$. Next set $a=x〈x^{p^{\alpha -ϵ+i}}〉$ and $b=y〈x^{p^{\alpha -ϵ+i}}〉$ and observe that $H(i+j)/A_i\cong H(i).$ Finally, $\left|A_i\right|=p^{(\alpha -ϵ+i+j)-(\alpha -ϵ+i)}=p^j=o([\omega ])$ and the proof is completed as in that of Theorem 4.6.

In case (b) the argument is now essentially the same as in case (a), but with ϵ and γ interchanged. □

The group $G=G_p(\alpha ,\beta ,ϵ,\gamma )$ is obtained in cases (a) and (b) in Theorem 4.7 by setting $i=\alpha -\gamma$ and $\alpha -ϵ$ respectively and the definitive choice we make for an ω_j-covering group for G is $H(\alpha -\gamma +j)$ and $H(\alpha -ϵ+j)$ respectively, where $[{\omega }_j]$ has order p^j and $0\le j\le \alpha -(ϵ+\gamma ).$

5 Regular elements and projective characters of metacyclic p-groups of positive type

We first aim to find the ω_i-regular elements of a metacyclic p-group G of positive type, where ω_i is a 2-cocycle of G with $o([{\omega }_i])=p^i$ in M(G). The first result is a slight generalization of [10, Lemma 4]. The original result is stated for the case when ϵ = 0, but the proof of Theorem 4 of the same paper shows that this result holds for any metacyclic p-group of positive type.

Lemma 5.1.

Let $H=G_p(\alpha ,\beta ,ϵ,\gamma )=〈x,y〉.$ Set $z=x^{p^{\alpha -1}}$ and let $G=H/〈z〉$. Then $h〈z〉,h\prime 〈z〉\in G-Z(G)$ are conjugate in G if and only if h and $h\prime$ are conjugate in H.

For notational convenience we now reset the indices in Theorems 4.6 and 4.7, so that $H(0)=G_p(\alpha ,\beta ,ϵ,\gamma ).$

Corollary 5.2.

Let $G\cong G_p(\alpha ,\beta ,ϵ,\gamma )$ with M(G) non-trivial and let $[\omega ]\in M(G)$ with $[\omega ]\ne [1].$ Then all the ω-regular elements of G are central.

Proof.

Using (the now revised indices) we set $H(1)=〈x,y〉=G_p(\alpha +1,\beta ,ϵ,\gamma +1)$ in Theorem 4.6 and $G_p(\alpha +1,\beta ,ϵ+1,\gamma +1)$ in Theorem 4.7. Then from the two theorems $A_1=〈z〉=〈x^{p^{\alpha }}〉$ and H(1) is an ω₁-covering group for G, where $[{\omega }_1]\in M(G)$ with $o([{\omega }_1])=p.$

Now Lemma 5.1 yields that $h〈z〉,h\prime 〈z〉\in G-Z(G)$ are conjugate in G if and only if h and $h\prime$ are conjugate in H(1). In particular, h is conjugate to hz^k in H(1) for all $k\in \mathbb{N}$ and for all $h〈z〉\in G-Z(G)$. Next, if we take any $\chi \in \mathit{Irr}(H(1)|\lambda )$ with $\text{tra}(\lambda )=[{\omega }_1],$ then $\lambda (z)=e^{2\pi ik/p}$ for some $k\in \{1,\dots ,p-1\}$. However this means that $\chi (h)=\chi (hz)=\chi (h)\lambda (z)$. In particular, since $\lambda (z)\ne 1$, we must have $\chi (h)=0$, and thus $h〈z〉$ is not ω₁-regular. Then since M(G) is cyclic every non-trivial $[\delta ]\in M(G)$ satisfies ${[\delta ]}^n=[{\omega }_1]$ for some $n\in \mathbb{N}$. Therefore, $G_{\delta }\subseteq G_{{\delta }^n}=G_{{\omega }_1}$ by Lemma 2.3 and so the result for $[{\omega }_1]$ extends to all nontrivial elements of M(G). □

Corollary 5.3.

Let $G\cong G_p(\alpha ,\beta ,ϵ,\gamma )$ and let $[{\omega }_i]\in M(G)$ of order pⁱ for $i=0,\dots ,v_p(|M(G)|).$ Then $Z_{{\omega }_i}(G)\cong 〈a^{p^{\gamma +i}},b^{p^{\gamma +i}}〉$.

Proof.

Let $H(i)=〈x,y〉$ be the ω_i-covering group of $G=H(0)$ (with indices reset) as given in Theorems 4.6 and 4.7. Then by [12, Theorem 3.5], $Z(H(i))/〈x^{p^{\alpha }}〉\cong Z_{{\omega }_i}(G)$. Furthermore $Z(H(i))=〈x^{p^{\gamma +i}},y^{p^{\gamma +i}}〉$ from Lemma 4.5. So writing $A_i=〈x^{p^{\alpha }}〉$ we obtain $$Z_{{\omega }_i}(G)\cong 〈x^{p^{\gamma +i}},y^{p^{\gamma +i}}〉/A_i=〈x^{p^{\gamma +i}}{A}_i,y^{p^{\gamma +i}}{A}_i〉\cong 〈a^{p^{\gamma +i}},b^{p^{\gamma +i}}〉.$$

Our second aim in this section is to describe the irreducible ω_i-characters of $G=G_p(\alpha ,\beta ,ϵ,\gamma )$ for the 2-cocycles ω_i described below. This will be achieved by describing the elements of $\text{Proj}(Z_i,{({\omega }_i)}_{Z_i}),$ where $Z_i=Z_{{\omega }_i}(G),$ and by finding the degrees of the elements of $\text{Proj}(G,{\omega }_i).$

To achieve the first aim we need to choose a transversal $R=\{r(g):g\in G\}$ for A_i in H(i), where A_i and $H(i)=〈x,y〉$ are as described in Theorems 4.6 and 4.7 with indices reset so that $H(0)=G.$ A natural choice for the elements of R in $Z(H(i))$ is to define $r(a^{p^{\gamma +i}s}{b}^{p^{\gamma +i}t})=x^{p^{\gamma +i}s}{y}^{p^{\gamma +i}t}$ for all $s\in \{0,\dots ,p^{\alpha -(\gamma +i)}-1\}$ and for all $t\in \{0,\dots ,p^{\beta -(\gamma +i)}-1\}$. Now let λ_i be the element of $\mathit{Irr}(A_i)$ with ${\text{tra}}_i({\lambda }_i)={\omega }_i$ so that both ω_i and $[{\omega }_i]$ have order pⁱ as explained in Section 1. Define ${\zeta }_i={\lambda }_i(x^{p^{\alpha }}),$ so that ζ_i is a primitive pⁱth root of unity.

Lemma 5.4.

With notation as above $\text{Proj}(Z_i,{({\omega }_i)}_{Z_i})=\{\delta \mu :\mu \in \mathit{Irr}(Z_i)\}$ with $$\delta (a^{p^{\gamma +i}s}{b}^{p^{\gamma +i}t})={\zeta }^s,$$for $s\in \{0,\dots ,p^{\alpha -(\gamma +i)}-1\},t\in \{0,\dots ,p^{\beta -(\gamma +i)}-1\},$ and where ζ is a $p^{\alpha -\gamma }$th root of unity with ${\zeta }^{p^{\alpha -(\gamma +i)}}={\zeta }_i.$

Proof.

Let T_i be the subgroup of $Z(H(i))$ which is mapped to Z_i under the epimorphism ${\pi }_i:H(i)\to G$ with kernel A_i. Then $T_i=〈x^{p^{\gamma +i}},y^{p^{\gamma +i}}〉\cong C_{p^{\alpha -(\gamma +i)}}\times C_{p^{\beta -(\gamma +i)}}.$ So we just need to find one $\eta \in \mathit{Irr}(T_i)$ satisfying $\eta {(x^{p^{\gamma +i}})}^{p^{\alpha -(\gamma +i)}}=\eta (x^{p^{\alpha }})={\zeta }_i.$ One such character η is defined by $\eta (x^{p^{\gamma +i}})=\zeta ,$ where ${\zeta }^{p^{\alpha -(\gamma +i)}}={\zeta }_i$ and $\eta (y^{p^{\gamma +i}})=1.$ Finally δ can be constructed by defining $\delta (a^{p^{\gamma +i}s}{b}^{p^{\gamma +i}t})=\eta (r(a^{p^{\gamma +i}s}{b}^{p^{\gamma +i}t}))=\eta (x^{p^{\gamma +i}s}{y}^{p^{\gamma +i}t})={\zeta }^s.$

Finally $\text{Proj}(Z_i,{({\omega }_i)}_{Z_i})=\{\delta \mu :\mu \in \mathit{Irr}(Z_i)\}.$ However, $Z_i=〈a^{p^{\gamma +i}},b^{p^{\gamma +i}}〉\cong C_p^{p^{\alpha -(\gamma +i)}-1}\times C_p^{p^{\beta -(\gamma +i)}-1}.$ □

We now observe from Corollary 5.2 and Proposition 2.2, that $\xi {(1)}^2=|G:Z_i|=p^{2(\gamma +i-1)}$ for all $\xi \in \text{Proj}(G,{\omega }_i)$ for $i\ne 0.$ Secondly we have $\xi (g)=0$ for all $g\notin Z_i$ and all $\xi \in \text{Proj}(G,{\omega }_i)$ for $i\ne 0.$ Thirdly there is a bijection between the elements of $\text{Proj}(G,{\omega }_i)$ and the elements of $\mathit{Irr}(Z_i)$ defined by $\xi \mapsto \mu$ if $\xi (g)=\xi (1)\delta (g)\mu (g)$ for all $g\in Z_i,$ where δ is as described in Lemma 5.4 and $i\ne 0.$ We thus have a complete description of the elements of $\text{Proj}(G,{\omega }_i)$ for $i\ne 0;$ however, the exact values of δ depend upon the initial choice of representation group that we made and the consequent choice of transversal R. The differences these choices make are discussed in detail in [4], but in essence it means that δ is only defined up to certain roots of unity.

6 Metacyclic 2-groups of negative type

Let G be a metacyclic 2-group of negative type. In this section we will find $M(G),$ a representation group for G that is a Schur group, $|\text{Proj}(G,\omega )|$ and the degree of each element of $\text{Proj}(G,\omega )$ for $[\omega ]$ non-trivial in $M(G).$

We first compute the Schur multipliers of all groups of negative type.

Proposition 6.1.

Let $G\cong G_2(\alpha ,\beta ,ϵ,\gamma ,-).$ Then

1. if ϵ = 0 and $\beta >\gamma ,$ then $M(G)\cong C_2;$

2. if $ϵ=1,$ or ϵ = 0 and $\beta =\gamma ,$ then M(G) is trivial.

Proof.

We use Theorem 4.1 and its notation. Converting notations, we have $m=2^{\alpha },s=2^{\beta },r=2^{\alpha -\gamma }-1,$ and $t=2^{\alpha -ϵ}.$ We see that $(r-1,m)=(2^{\alpha -\gamma }-2,2^{\alpha })=2,v_2(1+r+\cdots +r^{s-1})=\alpha -\gamma +\beta -1$ from [1, Lemma 2.3] and $\alpha -\gamma +\beta -1\ge \alpha -1$ with equality if and only if $\beta =\gamma .$ Now overall we have $n=2l/2^{\alpha }=l/2^{\alpha -1}$, where $l=(2^{\alpha -\gamma +\beta -1},2^{\alpha -ϵ}).$ Now if ϵ = 0 and $\beta >\gamma ,$ then $l=2^{\alpha }.$ On the other hand if $ϵ=1,$ or ϵ = 0 and $\beta =\gamma ,$ then $l=2^{\alpha -1.}$ □

We next aim to find a representation group, that is also a Schur group, for $G\cong G_2(\alpha ,\beta ,ϵ,\gamma ,-)$ when G has nontrivial Schur multiplier, that is when ϵ = 0 and $\beta >\gamma .$ One method of constructing such a representation group, that is also a Schur group, can be seen in the proof of [3, Theorem IV.2.10]. More specifically, we start with any metacyclic group $$G=〈a,b:a^m=1,b^s=a^t,ba{b}^{-1}=a^r〉,$$where $k=\lambda m/(m,r-1)$ (with λ as defined in Section 3). We then find an integer u satisfying $(u,m)=1$ and $u\equiv (r-1)/(m,r-1)\hspace{1em}(\mathrm{mod}m/(m,r-1))$ and set $v\equiv u(m,r-1)+1\hspace{1em}(\mathrm{mod}m\lambda )$ with $v\in \mathbb{N}$ and $v\le m\lambda +1.$ Then a representation group for G is given by $$H=〈x,y:x^{m\lambda }=1,y^s=x^w,yx{y}^{-1}=x^v〉,$$ where $w=m\lambda /(m\lambda ,r-1)$.

Theorem 6.2.

Let $G\cong G_2(\alpha ,\beta ,0,\gamma ,-)$ with $\beta >\gamma .$ Then $H=G_2(\alpha +1,\beta ,1,\gamma +1,-)$ is a representation group for G and a Schur group.

Proof.

Using the notation above $m=2^{\alpha },s=2^{\beta },r=2^{\alpha -\gamma }-1,\lambda =2,m/(m,r-1)=2^{\alpha -1}$ and $(r-1)/(m,r-1)=2^{\alpha -\gamma -1}-1$. Set $u=2^{\alpha -\gamma -1}-1,v=2^{\alpha -\gamma }-1$ and $w=2^{\alpha }$ to obtain the representation group and Schur group $$H=〈x,y:x^{2^{\alpha +1}}=1,y^{2^{\beta }}=x^{2^{\alpha }},yx{y}^{-1}=x^{2^{\alpha -\gamma }-1}〉=G_2(\alpha +1,\beta ,1,\gamma +1,-).$$

The remainder of this section will be concerned with finding the degrees of the irreducible ω-characters of G, where $[\omega ]\ne [1]$. We will also see why the approach of Section 4 does not work for metacyclic 2-groups that are only of negative type. Both of these aims will be achieved through finding the number of irreducible ω-characters of G.

Now we have shown that every metacyclic 2-group G of negative type with non-trivial Schur multiplier has $M(G)\cong C_2$. So for any representation group H of G, $\mathit{Irr}(H)=\mathit{Irr}(H|{\lambda }_1)\dot{\cup }Irr(H|{\lambda }_2)$, where λ₁ and λ₂ are the trivial and nontrivial character of C₂ respectively. Thus, $|\mathit{Irr}(H|{\lambda }_1)|+|\mathit{Irr}(H|{\lambda }_2)|=k(H)$, the number of conjugacy classes of H. However, $|\mathit{Irr}(H|{\lambda }_1)|=|\mathit{Irr}(G)|=k(G)$, and therefore if ω is a 2-cocycle of G with $[\omega ]\ne [1]$, then $|\text{Proj}(G,\omega )|=|\mathit{Irr}(H|{\lambda }_2)|=k(H)-k(G)$. These numbers can mainly be obtained from [11, Theorems 3.1 and 3.4]. For convenience, we now summarize these results.

Lemma 6.3.

Let $G\cong G_2(\alpha ,\beta ,ϵ,\gamma ,-).$ Then $$k(G)=(2^{\gamma }-1)2^{\alpha +\beta -2\gamma -1}+2^{\alpha +\beta -\gamma -2}+3(2^{\beta -1}).$$

Proof.

For G of nilpotency class at least three this follows from the aforementioned result and [1, Proposition 3.7], since the same number is obtained for k(G) if γ = 0 or 1. We just have to check that the results hold when G has nilpotency class one or two. Now if G is abelian, we have that α = 1 and γ = 0 and so the formula for k(G) gives $2^{\beta +1}=\left|G\right|.$ Now for G of nilpotency class two we have that α = 2 and γ = 0, so that $|G/Z(G)|=2^2$ from [1, Proposition 2.5] and hence $k(G)=5(2^{\beta -1})=2^{\beta }+3(2^{\beta -1})$ from [10, Lemma 5]. □

Theorem 6.4.

Let $G\cong G_2(\alpha ,\beta ,0,\gamma ,-)$ with $\beta >\gamma .$ Let ω be a 2-cocycle of G with $[\omega ]\ne [1]$. Then $|\text{Proj}(G,\omega )|=2^{\alpha +\beta -2(\gamma +1)}$.

Proof.

We need to find $k(H)-k(G)$, where H is as described in Theorem 6.2. Now $$\begin{array}{c}k(H)-k(G)=(2^{\gamma +1}-1)2^{\alpha +\beta -2\gamma -2}-(2^{\gamma }-1)2^{\alpha +\beta -2\gamma -1}\\ =(2^{\gamma +1}-1)2^{\alpha +\beta -2\gamma -2}-(2^{\gamma +1}-2)2^{\alpha +\beta -2\gamma -2}\\ =2^{\alpha +\beta -2(\gamma +1)}.\end{array}$$

Proposition 6.5.

Let $G\cong G_2(\alpha ,\beta ,0,\gamma ,-)$ with $\beta >\gamma .$ Let ω be a 2-cocycle of G with $[\omega ]\ne [1]$. Then $Z_{\omega }(G)\cong 〈b^{2^{\gamma +1}}〉$.

Proof.

Let $H=〈x,y〉$ be the representation group for G defined in Theorem 6.2. Then by [1, Proposition 2.5] we have $Z(H)=Z(G_2(\alpha +1,\beta ,1,\gamma +1,-))=〈x^{2^{\alpha }},y^{2^{\gamma +1}}〉$. So from [12, Theorem 3.5] $$Z_{\omega }(G)\cong Z(H)/〈x^{2^{\alpha }}〉\cong 〈b^{2^{\gamma +1}}〉.$$

Note that if G is a non-abelian group in Proposition 6.5 then there are non-central ω-regular elements of G. To see this, by Theorem 6.4 the total number of ω-regular conjugacy classes of G is $2^{\alpha +\beta -2(\gamma +1)}$. However Proposition 6.5 yields that $2^{\beta }/2^{\gamma +1}=2^{\beta -(\gamma +1)}$ of these ω-regular conjugacy classes are central. This leaves a total of $2^{\alpha +\beta -2(\gamma +1)}-2^{\beta -(\gamma +1)}=2^{\beta -(\gamma +1)}(2^{\alpha -(\gamma +1)}-1)$ ω-regular conjugacy classes that are non-central. This number is nonzero since $\alpha >\gamma +1$ for all non-abelian metacyclic 2-groups of negative type. This observation implies that a metacyclic 2-group of nilpotency class two that is of both positive and negative type must have trivial Schur multiplier. Thus, this also offers an alternative proof that the groups $G_2(2,\beta ,0,0,-)$ for $\beta \in \mathbb{N}$ are of negative type only.

Theorem 6.6.

Let $G\cong G_2(\alpha ,\beta ,0,\gamma ,-)$ with $\beta >\gamma .$ Let ω be a 2-cocycle of G with $[\omega ]\ne [1]$. Then $\xi (1)=2^{\gamma +1}$ for all $\xi \in \text{Proj}(G,\omega )$.

Proof.

$H=G_2(\alpha +1,\beta ,1,\gamma +1,-)$ is a representation group for G by Theorem 6.2. Let $N=〈x^2,y^{2^{\gamma }}〉,$ so that N has index $2^{\gamma +1}$ in H. Now $$y^{2^{\gamma }}{x}^2{y}^{-2^{\gamma }}=x^{2{(2^{\alpha -\gamma }-1)}^{2^{\gamma }}}$$and $v_2({(2^{\alpha -\gamma }-1)}^{2^{\gamma }}-1)=\alpha$ from [1, Lemma 2.3], so that $2{(2^{\alpha -\gamma }-1)}^{2^{\gamma }}\equiv 2\hspace{1em}(\mathrm{mod}{2}^{\alpha +1})$ and hence N is an abelian subgroup of H.

From the presentation of H, it is clear that $y{x}^2{y}^{-1}\in N$. We also have $$x^{-1}{y}^{2^{\gamma }}x=x^{{(2^{\alpha -\gamma }-1)}^{2^{\gamma }}-1}{y}^{2^{\gamma }}\in N,$$and so N is a normal subgroup of H. Now $\chi (1)$ divides $|H:N|$ for all $\chi \in \mathit{Irr}(H)$ by Ito’s theorem (see [5, Theorem 6.5.6]). So in particular $\xi (1)$ divides $2^{\gamma +1}$ for all $\xi \in \text{Proj}(G,\omega ).$ Now using Theorem 6.4 $$\left|G\right|=2^{\alpha +\beta }={(2^{\gamma +1})}^2|\text{Proj}(G,\omega )|=\sum _{\xi \in \text{Proj}(G,\omega )}\xi {(1)}^2.$$

Thus, the only possibility is $\xi (1)=2^{\gamma +1}$ for all $\xi \in \text{Proj}(G,\omega )$. □