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Abstract
Let ω be a 2-cocycle of a metacyclic p-group G representing a non-trivial element of the Schur multiplier 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:
2020 Mathematics Subject Classification:
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
for some
where Cn 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 so that
and
for all x, y,
The set of all such 2-cocycles of G form a group under multiplication. Two 2-cocycles ω and ψ are cohomologous if there exists a function
such that
where
for all
This defines an equivalence relation on
and the cohomology classes
form a finite abelian group, called the Schur multiplier
Definition 1.1.
Let ω be a 2-cocycle of a group G. Then is ω-regular if
for all
Let be the set of ω-regular elements of G. It is easy to check that if
then
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
then it does not follow in general that
We next define where Z(G) denotes the center of G. This set was studied in [12, Section 3] and in particular
is a subgroup of G. A further refinement of this concept is the epicenter
of G, which is defined by
The group G is said to be unicentral if 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 of invertible n × n matrices over
Definition 1.3.
Let ω be a 2-cocycle of a group G. Then is an ω-representation of G if
for all
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 for all
where P1 and P2 are also ω-representations of G. The set of irreducible ω-characters is denoted by
(or
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
if and only if
for all
or equivalently that
if and only if there exists
such that
(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 such that the order of ω equals the order of
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
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 such that
and
Details of the following basic concepts may be found in [Citation7] or [Citation4]. A representative class-preserving 2-cocycle ω for each can be constructed from a representation group H of G using each linear character
and the order of ω is also equal to that of
. This is achieved using the transgression isomorphism
In fact
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
are linearized by
where the latter set denotes the set of
such that λ is a constituent of
It is also the case that
from [12, Theorem 3.5].
Given ω constructed from as above, we set
and
Then
is called an ω-covering group for G and ω in the sense that ω may be constructed from (with the obvious notation)
and
are linearized by
Again it is the case that
this time
defines an isomorphism from
to
In Section 2 we will consider groups G for which and show that the elements of
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
In Sections 4 and 5 the Schur multiplier, a representation group, the ω-regular elements and the corresponding projective character tables for all
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,
and the degree of each element of
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, is defined by
where
for all
For notational convenience we will also define
in this context.
Lemma 2.1.
Let ω be a 2-cocycle of a group G, let and let
Then
is trivial;
is in the image of
if
, then
for some
if and only if
for all
if and only if
for some
if
then there exists
with
such that
for
and some
, where if
then
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 divides
for all
and
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 This means that the corresponding projective character table of G has just one single entry of
We also note in passing that if
is trivial, then
for
with
Proposition 2.2.
Let ω be a 2-cocycle of a group G such that Then, with
for all
for some
and G/U is of
-central type for all such
Proof.
Using the notation of Lemma 2.1 (f), and so
for all
and hence G/U is of
-central type. Finally
□
We may vary Proposition 2.2 by replacing ω by as shown in Lemma 2.1 (f), so that
The price for doing this is that the order of the original 2-cocycle ω now divides that of
and
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
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 and
for
and let
denote the inflation homomorphism from
and
respectively into
Then
there exists
and
such that
and
Proof.
For (a), it is obvious that if is ω-regular, then it is also ωn-regular.
For (b), from Lemma 2.1(b) there exists such that
Now
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 -regular Thus from Lemma 2.1(b) there exists
with
Finally, identifying the group
with G/V yields the desired result. □
In the context of Lemma 2.3 we have shown that we may replace ω by but it may not be the case that we can also replace ωn by
3 Metacyclic p-groups
Metacyclic groups G have presentations of the form
where m, r, s, and t are positive integers satisfying
and
Conversely if the positive integers
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
where
which means that we can, and will, assume that
Now Beyl, (see [3, pp. 194–195]), defines
and
, so that G has parameters m, r, s and
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 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:
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 and ϵ a non-negative integer with
such that for an odd prime number p,
whereas for
for
or
We note that, using the exponent of G and the classification of abelian groups, we have for G in Lemma 3.1 and that
and for
that
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
for
The next result is now obvious from these remarks.
Lemma 3.2.
Let and let ϵ be a non-negative integer with
Then
for p odd,
is a non-cyclic abelian metacyclic p-group of order
for
is a non-cyclic abelian metacyclic 2-group of order
for
and
is a non-cyclic abelian metacyclic 2-group of order
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 γ, and ϵ with
and
such that for an odd prime number p,
If then in addition
and
where in the second case Moreover, if
then
for all prime numbers p.
A non-cyclic metacyclic p-group G is of positive type if for any prime number p and of negative type if p = 2 and
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 ϵ and γ in Theorem 3.3 are satisfied then
defines a metacyclic p-group. The proof utilizes the p-adic valuation
of an integer.
Lemma 3.4.
Let γ and ϵ be non-negative integers with
and
Then
is a non-abelian metacyclic group of order
if
and if in addition for
is a non-abelian metacyclic group of order
if
and
Proof.
We first note that so
also if
then
so
Now we show that the conditions ensure that
and
in case (a) or
in case (b).
First we observe that so in case (a)
and this is divisible by
On the other hand in case (b),
and
so
and this is divisible by
Now
and for
so
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, is the only group in Lemma 3.1 that is only of negative type, whereas
for
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,
and
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: for
and
or
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:
where
with
and
for
the quaternion group of order 8.
Notice that this result implies that the groups for
are of both positive and negative type.
Lemma 3.7.
The groups for
and
are not of positive type.
Proof.
Let or
Then
and so if N is a cyclic normal subgroup of G with G/N cyclic, then
Now suppose with
Then
from [1, Proposition 2.5] and
Suppose N is a cyclic normal subgroup of G with
and G/N cyclic. Let
of order 8. Then
but
and we conclude that
Now from [1, Lemma 2.4],
where
but
from [1, Lemma 2.3]. Thus
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 [Citation1] and that of Beyl: and
. For groups of positive type, this leads to
and
. For groups of negative type we know that p = 2 and
. Therefore
and
.
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 to
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
where the positive integers m, r, s and t satisfy
Then where
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 Then
Proof.
Applying the conversion described in Section 3 to Theorem 4.1 we obtain that where
with
. Now
from [1, Lemma 2.3] and so
Beyl [Citation2] 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 Then G is a Schur group if
for
or
for
The non-isomorphic metacyclic p-groups of positive type that are Schur groups are
and
for
Proof.
The first part is immediate from Corollary 4.2 once we note that if then
from [1, Proposition 3.6]. For the second part if
and
then
On the other hand if
then
with equality if and only if
In the latter case we obtain
again from [1, Proposition 3.6]. Finally the groups
and
for
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 Then
and
.
Theorem 4.6.
Let with
Define the sequence
for
if
for
if
Then H(i) is a metacyclic p-group and in case (a) and
in case (b). Suppose
of order
Then
is an ω-covering group for
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), By definition
also
and
Thus, H(i) is a metacyclic p-group from Lemma 3.2,
from Corollary 4.2 and in particular
is a Schur group.
Let Then by Lemma 4.5,
and
. Therefore,
, where
. Thus,
. Next set
and
and observe that
Finally,
and if we let
be the transgression homomorphism, then
since
is cyclic.
In case (b), and
Thus, H(i) is a metacyclic p-group from Section 3,
from Corollary 4.2 and in particular
is a Schur group.
Using the same notation as in case (a), we have by Lemma 4.5 that and
. Therefore,
Thus
. Next set
and
and observe that
Finally,
and the subsequent argument is the same as in case (a). □
Notice in Theorem 4.6 that 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
and γ, respectively and the definitive choice we make for an ωj-covering group for G is
and
, respectively, where
has order pj and
Theorem 4.7.
Let with
Define the sequence
for
if
for
if
Then H(i) is a metacyclic p-group and in case (a) and
in case (b). Suppose
of order
Then
is an ω-covering group for
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), By definition
also
and
Thus H(i) is a metacyclic p-group from Lemma 3.2,
from Corollary 4.2 and in particular
is a Schur group.
Now let Then by Lemma 4.5,
and
. Therefore,
, where
. Thus,
. Next set
and
and observe that
Finally,
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 is obtained in cases (a) and (b) in Theorem 4.7 by setting
and
respectively and the definitive choice we make for an ωj-covering group for G is
and
respectively, where
has order pj and
5 Regular elements and projective characters of metacyclic p-groups of positive type
We first aim to find the ωi-regular elements of a metacyclic p-group G of positive type, where ωi is a 2-cocycle of G with 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 Set
and let
. Then
are conjugate in G if and only if h and
are conjugate in H.
For notational convenience we now reset the indices in Theorems 4.6 and 4.7, so that
Corollary 5.2.
Let with M(G) non-trivial and let
with
Then all the ω-regular elements of G are central.
Proof.
Using (the now revised indices) we set in Theorem 4.6 and
in Theorem 4.7. Then from the two theorems
and H(1) is an ω1-covering group for G, where
with
Now Lemma 5.1 yields that are conjugate in G if and only if h and
are conjugate in H(1). In particular, h is conjugate to hzk in H(1) for all
and for all
. Next, if we take any
with
then
for some
. However this means that
. In particular, since
, we must have
, and thus
is not ω1-regular. Then since M(G) is cyclic every non-trivial
satisfies
for some
. Therefore,
by Lemma 2.3 and so the result for
extends to all nontrivial elements of M(G). □
Corollary 5.3.
Let and let
of order pi for
Then
.
Proof.
Let be the ωi-covering group of
(with indices reset) as given in Theorems 4.6 and 4.7. Then by [12, Theorem 3.5],
. Furthermore
from Lemma 4.5. So writing
we obtain
Our second aim in this section is to describe the irreducible ωi-characters of for the 2-cocycles ωi described below. This will be achieved by describing the elements of
where
and by finding the degrees of the elements of
To achieve the first aim we need to choose a transversal for Ai in H(i), where Ai and
are as described in Theorems 4.6 and 4.7 with indices reset so that
A natural choice for the elements of R in
is to define
for all
and for all
. Now let λi be the element of
with
so that both ωi and
have order pi as explained in Section 1. Define
so that ζi is a primitive pith root of unity.
Lemma 5.4.
With notation as above with
for
and where ζ is a
th root of unity with
Proof.
Let Ti be the subgroup of which is mapped to Zi under the epimorphism
with kernel Ai. Then
So we just need to find one
satisfying
One such character η is defined by
where
and
Finally δ can be constructed by defining
Finally However,
□
We now observe from Corollary 5.2 and Proposition 2.2, that for all
for
Secondly we have
for all
and all
for
Thirdly there is a bijection between the elements of
and the elements of
defined by
if
for all
where δ is as described in Lemma 5.4 and
We thus have a complete description of the elements of
for
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 [Citation4], 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 a representation group for G that is a Schur group,
and the degree of each element of
for
non-trivial in
We first compute the Schur multipliers of all groups of negative type.
Proposition 6.1.
Let Then
if ϵ = 0 and
then
if
or ϵ = 0 and
then M(G) is trivial.
Proof.
We use Theorem 4.1 and its notation. Converting notations, we have and
We see that
from [1, Lemma 2.3] and
with equality if and only if
Now overall we have
, where
Now if ϵ = 0 and
then
On the other hand if
or ϵ = 0 and
then
□
We next aim to find a representation group, that is also a Schur group, for when G has nontrivial Schur multiplier, that is when ϵ = 0 and
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
where
(with λ as defined in Section 3). We then find an integer u satisfying
and
and set
with
and
Then a representation group for G is given by
where
.
Theorem 6.2.
Let with
Then
is a representation group for G and a Schur group.
Proof.
Using the notation above and
. Set
and
to obtain the representation group and Schur group
The remainder of this section will be concerned with finding the degrees of the irreducible ω-characters of G, where . 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 . So for any representation group H of G,
, where λ1 and λ2 are the trivial and nontrivial character of C2 respectively. Thus,
, the number of conjugacy classes of H. However,
, and therefore if ω is a 2-cocycle of G with
, then
. 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 Then
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 Now for G of nilpotency class two we have that α = 2 and γ = 0, so that
from [1, Proposition 2.5] and hence
from [10, Lemma 5]. □
Theorem 6.4.
Let with
Let ω be a 2-cocycle of G with
. Then
.
Proof.
We need to find , where H is as described in Theorem 6.2. Now
Proposition 6.5.
Let with
Let ω be a 2-cocycle of G with
. Then
.
Proof.
Let be the representation group for G defined in Theorem 6.2. Then by [1, Proposition 2.5] we have
. So from [12, Theorem 3.5]
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 . However Proposition 6.5 yields that
of these ω-regular conjugacy classes are central. This leaves a total of
ω-regular conjugacy classes that are non-central. This number is nonzero since
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
for
are of negative type only.
Theorem 6.6.
Let with
Let ω be a 2-cocycle of G with
. Then
for all
.
Proof.
is a representation group for G by Theorem 6.2. Let
so that N has index
in H. Now
and
from [1, Lemma 2.3], so that
and hence N is an abelian subgroup of H.
From the presentation of H, it is clear that . We also have
and so N is a normal subgroup of H. Now
divides
for all
by Ito’s theorem (see [5, Theorem 6.5.6]). So in particular
divides
for all
Now using Theorem 6.4
Thus, the only possibility is for all
. □
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