Rings of congruence preserving functions, II

Abstract

For several classes of special p-groups G, of exponent p, p > 2, we show that the near-ring, $C_0(G),$ of congruence preserving functions on G is a ring if and only if G is a 1-affine complete group.

Keywords:

• 1-Affine complete groups
• congruence preserving functions
• special p-groups

2010 Mathematics Subject Classification::

• 0A30
• 16Y30
• 20D15

1. Introduction

Let $G:=〈G,+,0〉$ be a finite group written additively but not necessarily abelian, with neutral element 0. As usual, we let $M_0(G)$ denote the near-ring of zero-preserving functions on G under the operations of pointwise addition and function composition. We consider subnear-rings $P_0(G),$ the near-ring of polynomial functions on G, and $C_0(G),$ the near-ring of congruence preserving functions on G. We recall $P_0(G)$ is the subnear-ring generated by the inner automorphisms of G while a function $f\in M_0(G)$ is congruence preserving if, for each $x,y\in G$ and normal subgroup N of G, if $x-y\in N,$ then $f(x)-f(y)\in N.$

We let $\eta (G)$ denote the lattice of normal subgroups of G and recall $\eta (G)$ is lattice isomorphic to the congruence lattice of G. For any subgroup H of G, the normal closure $\overline{H}$ of H is defined by $\overline{H}=\cap \{N\in \eta (G)|N\supseteq H\}.$ For $x\in G$ we let $\overline{x}=\overline{〈x〉}$ and thus we have $f\in C_0(G)$ if and only if $f(x)-f(y)\in \overline{x-y}$ for all $x,y\in G.$ We have $I(G)=〈\text{Inn}(G)〉=P_0(G)\subseteq C_0(G)\subseteq M_0(G).$

In this paper we continue the investigation initiated in [11] as to when $C_0(G)$ is a ring. Of course, if $C_0(G)$ is a ring so is $P_0(G)$ and from Chandy ([3]), $P_0(G)$ is a ring if and only if G is a 2-Engel group, i.e., every element of G commutes with all of its conjugates. Since a group G of nilpotency class at most 2 is 2-Engel, in this investigation we restrict to nilpotent groups of class at most 2, and using standard results, can restrict to p-groups of class at most 2. (See [11].) For finite abelian groups, A, $C_0(A)$ is a ring if and only if A is 1-affine complete, and the 1-affine complete finite abelian groups are known ([11]). Recall that a group G is 1-affine complete (1-ac) means $C_0(G)=P_0(G).$ For background material and history see [10, pp 158–160].

Several necessary conditions on finite non-abelian nilpotent p-groups of class 2 for $C_0(G)$ to be a ring were given in [11] (see Theorem II.1 below) and in these cases for $p\ne 2$ all the groups G were 1-ac. The first examples of 1-ac non-abelian p-groups were given by Dorda ([6]). These groups were p-groups, nilpotent of class 2, $\text{exp}\hspace{0.17em}(G)=p,\left|G\right|=p^6,p>2.$

In light of this example and some GAP examples, we restrict our attention to finite non-abelian p-groups, G, $p\ne 2,$ of class 2 and $Z(G)=[G,G]$ and $G/[G,G]$ is elementary abelian, that is a special p-group. Recall that a finite group G is special if G is elementary abelian or G is nilpotent of class 2, $Z(G)=[G,G]$ and $G/[G,G]$ is elementary abelian. (The first occurrence we have found of these groups is in Hall and Higman ([9])). From group theory one finds that a non-abelian p-group, G, is special if G is nilpotent of class 2 and $Z(G)=[G,G]=\Phi (G)$ (the Frattini subgroup of G). A special p-group has exponent p or p² ([7]). We focus here on non-abelian special p-groups, G, $\text{exp}\hspace{0.17em}(G)=p,p\ne 2.$

A further reason for restricting to these special p-groups is that Verardi ([13]) has shown that there exists an injective map $G\mapsto G_p$ from the class of finite groups into the class of special p-groups of exponent p. Thus information about the associated special p-group G_p may be used to obtain information about G.

In the remainder of the paper G will denote a non-abelian special p-group, $p\ne 2,$ of exponent p. As usual, Z(G) denotes the center of G, $[G,G]=G\prime ,$ the commutator subgroup, and $M_0(G),C_0(G)$ and $P_0(G)$ as defined above.

2. Background results: old and new

As indicated at the end of the previous section, henceforth our groups G will be non-abelian special p-groups of exponent $p\ne 2.$ For ease of exposition we denote this by “Let $G\in \text{NAS}(\hspace{0.17em}\text{exp}\hspace{0.17em}p\ne 2)$”.

For $G\in \text{NAS}(\hspace{0.17em}\text{exp}\hspace{0.17em}p\ne 2),$ let $\eta (G)$ denote the lattice of normal subgroups of G. Let $D,E\in \eta (G),D⊊G,\left\{0\right\}⊊E.$ The pair (D, E) is called a splitting pair if for each $N\in \eta (G),N\subseteq D$ or $N\supseteq E.$ If G contains a splitting pair, we say G splits or G is split. In the case D = E, we say D is a cutting element and G is cut.

For $x\in G,$ we let $[x,G]=〈[x,g]|g\in G〉=\{[x,g]|g\in G\}.$ We have $\overline{x}=〈x〉+[x,G]$ and $\overline{x}$ is abelian ([11, 3.3]).

For use in the sequel we collect some (mostly) known results. We note that some of these hold for any non-abelian p-group of nilpotency class 2.

Theorem II.1.

Let $G\in \text{NAS}(\hspace{0.17em}\text{exp}\hspace{0.17em}p\ne 2)$. If any one of the following holds:

1. G is split ([11, 3.1]);

2. $\left|G\right|<p^6$ ([11, 4.7]);

3. G is abelian by cyclic ([11, 4.6]);

4. There exists $x\in G$ such that $[x,G]$ is cyclic ([6, Hilfsatz 9]);

5. The derived subgroup $G\prime =[G,G]$ is cyclic ([11, 4.1]);

6. $G\prime$ is 2-generated, that is $G\prime \cong {\mathbb{Z}}_p\oplus {\mathbb{Z}}_p,$

then $C_0(G)$ is not a ring and thus G is not 1-ac.

Proof.

For (4), Dorda ([6]) constructs a function $g\in C_0(G)-P_0(G).$ One finds that $g(id+id)\ne g·id+g·id$ so $C_0(G)$ is not a ring. For (6), we take $G\prime =〈c_1,c_2〉\cong {\mathbb{Z}}_p\oplus {\mathbb{Z}}_p.$ Let $x\in G-Z(G)$ and note $[x,G]=G\prime ,$ otherwise $[x,G]$ is cyclic and the result follows from (4). For $N\in \eta (G),$ if $N⊈Z(G)$ then for $x\in N-Z(G),G\prime =[x,G]\subseteq N.$ Thus Z(G) cuts G and we use (1).□

We mention two additional cases. In [4], Corsi Tani proved that if G is a finite p-group of nilpotency class 2 having an automorphism $\sigma \ne id,$ with $\gcd (|\sigma |,p)=1,$ and such that $\sigma (N)\subseteq N$ for all $N\in \eta (G),$ then $G\in \text{NAS}(\hspace{0.17em}\text{exp}\hspace{0.17em}p\ne 2)$ and G is cut. Thus these groups are not 1-ac and $C_0(G)$ is not a ring. Gorenstein ([7]) calls $G\in \text{NAS}(\hspace{0.17em}\text{exp}\hspace{0.17em}p\ne 2)$ extra special if $|G\prime |=p$ and proceeds to discuss the use of extra special groups in the classification problem of finite simple groups. From Theorem II.1, extra special p-groups, G, are not 1-ac and $C_0(G)$ is not a ring.

We know if G is not cut then $\eta (G)$ is a simple lattice ([2, Lemma 6.1]). For $H,K\in \eta (G)$ the interval I(H, K) is said to be a prime interval if $|I(H,K)|=2$ and in this case we write $H\prec K.$ From lattice theory, when $\eta (G)$ is simple then any two prime intervals are projective, hence if $H\prec K$ and $A\prec B,$ then B/A and K/H are $C_0(G)$ isomorphic (See also [1] and [2]).

Lemma

II.2. Let $G\in \text{NAS}(\hspace{0.17em}\text{exp}\hspace{0.17em}p\ne 2),|Z(G)|\ge p^2$. Let $f\in C_0(G)$. Then

1. 1. $f{|}_{Z(G)}=k·id,k\in {\mathbb{Z}}_p.$

2. If G is not cut, then $h=f-k·id\in C_0(G)$, where $k\in {\mathbb{Z}}_p$ is given in part 1, and $h(G)\subseteq Z(G),h(Z(G))=\left\{0\right\}.$

Proof.

1. Let $x,y\in Z(G)$ and $f\in C_0(G).$ Then $f(x)\in \overline{x}$ so $f(x)=k_x·x$ and $f(y)=k_y·y,k_x,k_y\in {\mathbb{Z}}_p.$ Since $f\in C_0(G),f(x)-f(y)\in \overline{x-y},$ that is $k_x·x-k_y·y=k(x-y),k\in {\mathbb{Z}}_p.$ Therefore $(k_x-k)x=(k_y-k)y.$ If $〈x〉\ne 〈y〉,$ then $k_x=k=k_y.$ If $〈x〉=〈y〉,$ then since $|Z(G)|\ge p^2,$ there exists $w\in Z(G),〈w〉\ne 〈x〉$ and $〈w〉\ne 〈y〉.$ But then $k_x=k_y=k_w$ so $f{|}_{Z(G)}=\mathcal{\ell }·id,\mathcal{\ell }\in {\mathbb{Z}}_p.$ (This also follows from ([12]) since Z(G) is affine complete.)

2. Let $N,M\in \eta (G),\left\{0\right\}\prec N\subseteq Z(G)$ and $M\prec G.$ Then $I(\{0\},N)$ and I(M, G) are projective so G/M and $N/\left\{0\right\}$ are $C_0(G)$ isomorphic. For $f\in C_0(G)$ we find from the first part, $f{|}_{Z(G)}=\mathcal{\ell }·id,\mathcal{\ell }\in {\mathbb{Z}}_p,$ so for $h=f-\mathcal{\ell }·id,h(Z(G))=(f-\mathcal{\ell }·id)(Z(G))=\left\{0\right\}.$ Thus $h(G)\subseteq \cap \{M\in \eta (G)|M\prec G\}\subseteq Z(G).$□

From Theorem II.1 we know if G is cut, G is not 1-ac. Thus in the sequel, when G is not cut and when attempting to show that arbitrary $f\in C_0(G)$ is also a polynomial function, without loss of generality we consider $h=f-\mathcal{\ell }·id.$

We introduce some further notation and concepts. Let $G\in \text{NAS}(\hspace{0.17em}\text{exp}\hspace{0.17em}p\ne 2).$ Then $G/Z(G)$ and $Z(G)=G\prime$ are ${\mathbb{Z}}_p$-vector spaces, say $$G/Z(G)=〈e_1+Z(G),\dots ,e_n+Z(G)〉\mathrm{ }\text{and}\mathrm{ }Z(G)=〈c_1,\dots ,c_s〉$$ so $G=〈e_1,\dots ,e_n,c_1,\dots ,c_s〉$ and $\left|G\right|=p^{n+s},n={\dim }_{{\mathbb{Z}}_p}(G/Z(G)),s={\dim }_{{\mathbb{Z}}_p}(Z(G)).$ Note $T:=\{[e_i,e_j]|1\le i<j\le n\}$ is a generating set for $Z(G)=G\prime$ so without loss of generality we take $B:=\{c_1,\dots ,c_s\}\subseteq T.$ Thus each $[e_i,e_j]\in T-B$ is a linear combination of elements from B. See Cortini ([5]) for this representation of $G\in \text{NAS}(\hspace{0.17em}\text{exp}\hspace{0.17em}p\ne 2).$

We mention that another computational approach to non-abelian special p-groups of exponent p is given by Grundhöfer and Stroppel ([8]) in their investigations of Heisenberg groups. This approach is used to obtain information about automorphisms of these special p-groups.

We next introduce a directed graph in which the defining information of our groups is enclosed. Let $G\in \text{NAS}(\hspace{0.17em}\text{exp}\hspace{0.17em}p\ne 2)$ be given by $G=〈e_1,\dots ,e_n,c_1,\dots ,c_s〉$ and the linear combinations for $[e_{\mathcal{\ell }},e_k]\in T-B,[e_{\mathcal{\ell }},e_k]={\sum }_{i=1}^s{\alpha }_{l,k}^i{c}_i.$ The vertices are the generators $\{e_1,\dots ,e_n\}$ and the directed edges are $[e_i,e_j],i<j.$ For $x,y\in G,x={\sum }_{i=1}^n{a}_i{e}_i+z_1,y={\sum }_{j=1}^n{b}_j{e}_j+z_2,z_1,z_2\in Z(G),[x,y]$ can be determined from the graph.

Example II.3

1. G is full. G is isomorphic to $〈e_1,\dots ,e_n,c_1,\dots ,c_s〉,s=(\begin{array}{c}n\\ 2\end{array}).$ In this case B = T. For n = 4 we have

2. G is circular. G is isomorphic to $〈e_1,\dots ,e_n,c_1,\dots ,c_n〉,[e_i,e_{i+1}]=c_i$ where we take $e_{n+1}=e_1,$ and other $[e_{\mathcal{\ell }},e_k]=0.$ For n = 4 we have

3. Consider G given by

   

where $[e_i,e_{i+1}]=c_i,1\le i\le 4$ (with e₅ = e₁), and $[e_2,e_4]=c_1+c_2.$ Note $[e_2,e_4]=[e_1,e_2]+[e_2,e_3]$ so $[e_2,e_4-e_3+e_1]=0.$ Thus G is isomorphic to where $b_1=e_1,b_2=e_2,b_3=e_3,b_4=e_4-e_3+e_1.$ Therefore G is circular.

In the next section, with the aid of this graphical representation, we determine new classes of non-abelian special p-groups of exponent p which are 1-ac and new classes which are not 1-ac. In these latter classes, $C_0(G)$ is not a ring.

3. Main results

As usual, $G\in \text{NAS}(\hspace{0.17em}\text{exp}\hspace{0.17em}p\ne 2),\left|G\right|=p^{n+s},n={\dim }_{{\mathbb{Z}}_p}(G/Z(G)),s={\dim }_{{\mathbb{Z}}_p}(Z(G)).$

Theorem III.1.

(Full) Let $G=〈e_1,\dots ,e_n,c_1,\dots ,c_s〉,n\ge 3,s=(\begin{array}{c}n\\ 2\end{array})$. Then G is 1-ac.

Proof.

Let $f\in C_0(G).$ As we have shown above, we may assume that $f(G)\subseteq Z(G),$ so we let $f(u)=[u,d_u],u,d_u\in G.$ We also let $[e_i,e_j]=c_{ij}$ for $1\le i<j\le n.$ From $f(e_2)-f(e_1)\in [e_2-e_1,G]$ we have $f(e_2)-f(e_1)=[e_2-e_1,x]$ for some $x\in G.$ It follows that $$\begin{array}{c}[e_2,d_{e_2}-d_{e_1}]=[e_2,d_{e_2}]-[e_2,d_{e_1}]-[e_1,d_{e_1}]+[e_1,d_{e_1}]\\ =f(e_2)-f(e_1)+[e_1,d_{e_1}]-[e_2,d_{e_1}]\\ =[e_2-e_1,x]-[e_2-e_1,d_{e_1}]\\ \in [e_2-e_1,G].\end{array}$$

So we have $[e_2,d_{e_2}-d_{e_1}]\in [e_2-e_1,G]\cap [e_2,G].$ Furthermore, $$\begin{array}{c}[e_2-e_1,G]=〈[e_2-e_1,e_1],[e_2-e_1,e_2],[e_2-e_1,e_3],\dots ,[e_2-e_1,e_n]〉\\ =〈c_{12},c_{23}-c_{13},\dots ,c_{2n}-c_{1n}〉,\end{array}$$

while $$[e_2,G]=〈[e_2,e_1],[e_2,e_2],[e_2,e_3],\dots ,[e_2,e_n]〉=〈c_{12},c_{23},\dots ,c_{2n}〉.$$

From the linear independence of the c_ij, it follows that $[e_2-e_1,G]\cap [e_2,G]=〈c_{12}〉,$ and hence $d_{e_2}-d_{e_1}$ is forced to have the form $\alpha e_1+\beta e_2+c,\alpha ,\beta \in {\mathbb{Z}}_p,c\in Z(G).$ If we let $\widehat{d_1}=d_{e_1}+\alpha e_1+c$ then $[e_1,\widehat{d_1}]=[e_1,d_{e_1}]$ and $[e_2,d_{e_2}]=[e_2,d_{e_1}+\alpha e_1+\beta e_2+c]=[e_2,\widehat{d_1}+\beta e_2]=[e_2,\widehat{d_1}].$ We put $d=\widehat{d_1}$ so that $f(e_1)=[e_1,d]$ and $f(e_2)=[e_2,d].$ For $j\ge 3,$ it follows similarly that $f(e_j)-f(e_2)\in [e_j-e_2,G]$ and so $[e_j,d_{e_j}-d]\in [e_j-e_2,G]\cap [e_j,G]=〈c_{2j}〉.$ Using $f(e_j)-f(e_1)\in [e_j-e_1,G]$ we get $[e_j,d_{e_j}-d]\in [e_j-e_1,G]\cap [e_1,G]=〈c_{1j}〉$ so $[e_j,d_{e_j}-d]=0,$ and $f(e_i)=[e_i,d],i=1,2,\dots ,n.$

For $\alpha e_j,$ let $f(\alpha e_j)=[\alpha e_j,g],g\in G.$ Using $f(\alpha e_j)-f(e_i)\in [\alpha e_j-e_i,G],i\ne j,$ we get $[\alpha e_j,g-d]=0$ and $f(\alpha e_j)=[\alpha e_j,d].$

Thus we find $d\in G,f(\alpha e_i)=[\alpha e_i,d]$ for each $i\in \{1,\dots ,n\},\alpha \in {\mathbb{Z}}_p.$

Now let $u=u_i{e}_i+u_j{e}_j+c,i\ne j,u_i\ne 0\ne u_j,$ $c\in Z(G).$ As above we find $[u,d_u-d]\in [e_i,G]\cap [e_j,G]=〈c_{ij}〉$ and for $k\ne i,k\ne j,[u,d_u-d]\in [u+e_k,G]\cap [e_i,G]\cap [e_j,G]=\left\{0\right\}.$ Hence $f(u)=[u,d].$

By induction, $f({\sum }_{i=1}^n{u}_i{e}_i+c)=[{\sum }_{i=1}^n{u}_i{e}_i,d]$ so $f\in P_0(G),$ that is, G is 1-ac.

We may now assume $n\ge 4.$ For if n = 3, then $s\le (\begin{array}{c}3\\ 2\end{array})=3.$ If s = 3 then from the above theorem, G is 1-ac. If s = 1 or s = 2, then from Theorem II.1, $C_0(G)$ is not a ring. For n = 2, s = 1 we are again finished using Theorem II.1. It should be mentioned that the “full” case $n=3,s=3$ is the example of Dorda mentioned above.

Theorem III.2.

(Circular) Let G be circular, i.e., G is isomorphic to $〈e_1,\dots ,e_n,c_1,\dots ,c_n〉,[e_i,e_{i+1}]=c_i$ where we take $e_{n+1}=e_1$, and the other $[e_{\mathcal{\ell }},e_k]=0$. Then G is 1-ac if and only if $C_0(G)$ is a ring if and only if n is odd.

Proof.

We havewhere we take $c_i=[e_i,e_{i+1}]$ and identify $e_{n+1}=e_1,i=1,\dots ,n.$

Let n be even, $s=n,\left|G\right|=p^{2n}$ and define a function $h\in M_0(G)$ by $$h\left(\sum _{i=1}^n{k}_i{e}_i+c\right)=\sum _{i=1}^n{(-1)}^{i+1}{k}_i{k}_{i+1}{c}_i,\mathrm{ }\text{where}\mathrm{ }{k}_{n+1}=k_1,$$ and where $c\in Z(G)$ is arbitrary. We show $h\in C_0(G)$ and show $C_0(G)$ is not a ring.

Now $$\begin{array}{c}h\left(\sum _{i=1}^n{k}_i{e}_i\right)-h\left(\sum _{i=1}^n{\mathcal{\ell }}_i{e}_i\right)=\sum _{i=1}^n{(-1)}^{i+1}(k_i{k}_{i+1}-{\mathcal{\ell }}_i{\mathcal{\ell }}_{i+1})c_i,\mathrm{ }\text{where}\mathrm{ }{\mathcal{\ell }}_{n+1}={\mathcal{\ell }}_1\\ =(k_1{k}_2-{\mathcal{\ell }}_1{\mathcal{\ell }}_2)c_1-(k_2{k}_3-{\mathcal{\ell }}_2{\mathcal{\ell }}_3)c_2+(k_3{k}_4-{\mathcal{\ell }}_3{\mathcal{\ell }}_4)c_3-(k_4{k}_5-{\mathcal{\ell }}_4{\mathcal{\ell }}_5)c_4+-\cdots \\ +(k_{n-1}{k}_n-{\mathcal{\ell }}_{n-1}{\mathcal{\ell }}_n)c_{n-1}-(k_n{k}_1-{\mathcal{\ell }}_n{\mathcal{\ell }}_1)c_n.(*)\end{array}$$

Also, $$\begin{array}{c}[\sum _{i=1}^n{k}_i{e}_i-\sum _{i=1}^n{\mathcal{\ell }}_i{e}_i,G]=[\sum _{i=1}^n(k_i-{\mathcal{\ell }}_i)e_i,G]\\ =〈(k_n-{\mathcal{\ell }}_n)c_n-(k_2-{\mathcal{\ell }}_2)c_1,(k_1-{\mathcal{\ell }}_1)c_1-(k_3-{\mathcal{\ell }}_3)c_2,\\ (k_2-{\mathcal{\ell }}_2)c_2-(k_4-{\mathcal{\ell }}_4)c_3,(k_3-{\mathcal{\ell }}_3)c_3-(k_5-{\mathcal{\ell }}_5)c_4,\\ ⋮\hfill \\ (k_{n-2}-{\mathcal{\ell }}_{n-2})c_{n-2}-(k_n-{\mathcal{\ell }}_n)c_{n-1},(k_{n-1}-{\mathcal{\ell }}_{n-1})c_{n-1}-(k_1-{\mathcal{\ell }}_1)c_n〉.\end{array}$$

We see that $(*)$ can be written as $$\begin{array}{c}-k_1[(k_n-{\mathcal{\ell }}_n)c_n-(k_2-{\mathcal{\ell }}_2)c_1]+{\mathcal{\ell }}_2[(k_1-{\mathcal{\ell }}_1)c_1-(k_3-{\mathcal{\ell }}_3)c_2]\\ -k_3[(k_2-{\mathcal{\ell }}_2)c_2-(k_4-{\mathcal{\ell }}_4)c_3]+{\mathcal{\ell }}_4[(k_3-{\mathcal{\ell }}_3)c_3-(k_5-{\mathcal{\ell }}_5)c_4]\\ ⋮\hfill \\ -k_{n-1}[(k_{n-2}-{\mathcal{\ell }}_{n-2})c_{n-2}-(k_n-{\mathcal{\ell }}_n)c_{n-1}]+{\mathcal{\ell }}_n[(k_{n-1}-{\mathcal{\ell }}_{n-1})c_{n-1}-(k_1-{\mathcal{\ell }}_1)c_n],\end{array}$$ that is, $$h\left(\sum _{i=1}^n{k}_i{e}_i\right)-h\left(\sum _{i=1}^n{\mathcal{\ell }}_i{e}_i\right)\in \left[\sum _{i=1}^n(k_i-{\mathcal{\ell }}_i)e_i,G\right]$$ for all $k_i,{\mathcal{\ell }}_i\in {\mathbb{Z}}_p,$ which implies $h\in C_0(G).$ However, $h(e_1+e_2)=c_1=h(-e_1-e_2),$ from which it follows that $h°(-id)\ne -h,$ so $C_0(G)$ is not a ring.

Suppose now n is odd. As above we take $c_i=[e_i,e_{i+1}],1\le i\le n,$ and identify $e_{n+1}$ as e₁. We take $f\in C_0(G)$ and show $f\in P_0(G).$ Recall that we may assume without loss of generality that $f(x)\in [x,G]$ for all $x\in G.$

From $f(e_i)\in [e_i,G],1\le i\le n,$ we have $f(e_i)\in 〈c_{i-1},c_i〉,1\le i\le n,$ (identifying $c_0=c_n$), say $f(e_i)={\alpha }_{i,1}{c}_{i-1}+{\alpha }_{i,2}{c}_i,1\le i\le n,{\alpha }_{i,1},{\alpha }_{i,2}\in {\mathbb{Z}}_p.$ Next, from $f(e_i)-f(e_{i+2})\in [e_i-e_{i+2},G]$ we get $${\alpha }_{i,1}{c}_{i-1}+{\alpha }_{i,2}{c}_i-{\alpha }_{i+2,1}{c}_{i+1}-{\alpha }_{i+2,2}{c}_{i+2}={\lambda }_1(c_i+c_{i+1})+{\lambda }_2{c}_{i-1}+{\lambda }_3{c}_{i+2},{\lambda }_i\in {\mathbb{Z}}_p,$$ since $[e_i-e_{i+2},G]=〈c_i+c_{i+1},c_{i-1},c_{i+2}〉.$ This forces ${\alpha }_{i,2}=-{\alpha }_{i+2,1},1\le i\le n,$ and by putting ${\tau }_i={\alpha }_{i-1,2},1\le i\le n,$ we find that $f(e_i)=[e_i,d]$ where $d={\sum }_{i=1}^n{\tau }_i{e}_i.$

Take $p\in P_0(G)$ where $p(x)=[x,d],x\in G$ and let $h=f-p\in C_0(G).$ Now $h(e_i)=0,1\le i\le n,$ and we show h(x) = 0 for all $x\in G,$ that is $f\in P_0(G)$ and G is 1-ac.

Let $k\in {\mathbb{Z}}_p,k\notin \{0,1\}.$ Then $h(k{e}_i)-h(e_{i-2})\in [k{e}_i-e_{i-2},G]=〈c_{i-3},k{c}_{i-1}+c_{i-2},c_i〉,$ where we take $c_{-3}=c_{n-3},c_{-2}=c_{n-2},c_{-1}=c_{n-1}$ and $h(k{e}_i)-h(e_{i+2})\in [k{e}_i-e_{i+2},G]=〈c_{i-1},k{c}_i+c_{i+1},c_{i+2}〉$ which implies $h(k{e}_i)=0,k\notin \{0,1\}.$ But $h(k{e}_i)=0$ for $k\in \{0,1\}$ so we have $h(k{e}_i)=0,1\le i\le n,k\in {\mathbb{Z}}_p.$

We also find $h(k{e}_i+\mathcal{\ell }{e}_j)=0$ if $|i-j|\ge 2.$ In fact, for $|i-j|\ge 2,h(k{e}_i+\mathcal{\ell }{e}_j)-h(\mathcal{\ell }{e}_j)\in [e_i,G]=〈c_{i-1},c_i〉$ and $h(k{e}_i+\mathcal{\ell }{e}_j)-h(k{e}_i)\in [e_j,G]=〈c_{j-1},c_j〉.$ Since $〈c_{i-1},c_i〉\cap 〈c_{j-1},c_j〉=\left\{0\right\}$ for $|i-j|\ge 2,h(k{e}_i+\mathcal{\ell }{e}_j)=0,k,\mathcal{\ell }\in {\mathbb{Z}}_p.$

We next show that $h(k{e}_i+\mathcal{\ell }{e}_{i+1})=0,k,\mathcal{\ell }\in {\mathbb{Z}}_p-\left\{0\right\}.$ From $h(k{e}_i+\mathcal{\ell }{e}_{i+1})-h(\mathcal{\ell }{e}_{i+1})\in [e_i,G]=〈c_{i-1},c_i〉$ and $h(k{e}_i+\mathcal{\ell }{e}_{i+1})-h(k{e}_i)\in [e_{i+1},G]=〈c_i,c_{i+1}〉,$ we find that $h(k{e}_i+\mathcal{\ell }{e}_{i+1})\in 〈c_i〉,$ say $$h(k{e}_i+\mathcal{\ell }{e}_{i+1})={\rho }_i(k,\mathcal{\ell })c_i,\mathrm{ }\text{where}\mathrm{ }{\rho }_i(k,l)\in {\mathbb{Z}}_p.$$

But, for $m\in {\mathbb{Z}}_p-\left\{0\right\},h(k{e}_i+\mathcal{\ell }{e}_{i+1})-h(\mathcal{\ell }{e}_{i+1}+m{e}_{i+2})\in [k{e}_i-m{e}_{i+2},G]=〈k{c}_i+m{c}_{i+1},c_{i-1},c_{i+2}〉,$ that is, ${\rho }_i(k,\mathcal{\ell })c_i-{\rho }_{i+1}(\mathcal{\ell },m)c_{i+1}=\lambda (k{c}_i+m{c}_{i+1})$ for some $\lambda \in {\mathbb{Z}}_p.$ This implies that (1) $${\rho }_{i+1}(\mathcal{\ell },m)=-m{k}^{-1}{\rho }_i(k,\mathcal{\ell })$$(1) for all $1\le i\le n,k,\mathcal{\ell },m\in {\mathbb{Z}}_p-\left\{0\right\}.$ (Note that ${\rho }_i(k,\mathcal{\ell })=0$ if at least one of k and $\mathcal{\ell }$ is zero.) The right-hand side of (1) equals the left-hand side for all $k\in {\mathbb{Z}}_p-\left\{0\right\}.$ Hence, $k^{-1}{\rho }_i(k,\mathcal{\ell })={\rho }_i(1,\mathcal{\ell }),k\in {\mathbb{Z}}_p-\left\{0\right\},$ so that (2) $${\rho }_i(k,\mathcal{\ell })=k{\rho }_i(1,\mathcal{\ell }),k\in {\mathbb{Z}}_p-\left\{0\right\}.$$(2)

Hence, from (1), ${\rho }_{i+1}(\mathcal{\ell },m)=-m{\rho }_i(1,\mathcal{\ell }).$ Put $\mathcal{\ell }=1,$ and ${\rho }_{i+1}(1,m)=-m{\rho }_i(1,1),$ which implies $m^{-1}{\rho }_{i+1}(1,m)=-{\rho }_i(1,1),$ $m\in {\mathbb{Z}}_p-\left\{0\right\}.$ So, $m^{-1}{\rho }_{i+1}(1,m)={\rho }_{i+1}(1,1),$ giving (3) $${\rho }_{i+1}(1,m)=m{\rho }_{i+1}(1,1).$$(3)

From (2) and (3), (4) $${\rho }_i(k,\mathcal{\ell })=k\mathcal{\ell }{\rho }_i(1,1).$$(4)

Now, put $k=\mathcal{\ell }=m=1$ in (1). Then $${\rho }_1(1,1)=-{\rho }_n(1,1)={\rho }_{n-1}(1,1)=-{\rho }_{n-2}(1,1)=\cdots ={\rho }_2(1,1)=-{\rho }_1(1,1),$$ since n is odd. From ${\rho }_1(1,1)=-{\rho }_1(1,1)$ we have ${\rho }_1(1,1)=0,$ and hence also $-{\rho }_1(1,1)={\rho }_2(1,1)=\cdots =-{\rho }_n(1,1)=0.$ By (4), ${\rho }_i(k,\mathcal{\ell })=0,1\le i\le n,k,l\in {\mathbb{Z}}_p.$ This shows that $h(k{e}_i+\mathcal{\ell }{e}_{i+1})=0c_i=0,1\le i\le n,k,\mathcal{\ell }\in {\mathbb{Z}}_p.$ So we now have that $$h(k{e}_i+\mathcal{\ell }{e}_j)=0,1\le i,j\le n,k,\mathcal{\ell }\in {\mathbb{Z}}_p.$$

We proceed by induction. Let $2\le t<n,$ and assume that $h(k_1{e}_{i_1}+k_2{e}_{i_2}+\cdots +k_t{e}_{i_t})=0$ for any $\{i_1,i_2,\dots ,i_t\}\subset \{1,2,\dots ,n\}$ and any $k_1,k_2,\dots ,k_t\in {\mathbb{Z}}_p.$ Then, without loss of generality, put $w={\sum }_{i=1}^{t+1}{k}_i{e}_i,$ with all $k_i\ne 0.$ Then $h(w)-h(w-k_1{e}_1)\in [e_1,G]=〈c_n,c_1〉$ and $h(w)-h(w-k_3{e}_3)\in [e_3,G]=〈c_2,c_3〉$ implies $h(w)\in 〈c_n,c_1〉\cap 〈c_2,c_3〉=\left\{0\right\}.$ We conclude that $$h\left(\sum _{i=1}^n{k}_i{e}_i\right)=0,k_i\in {\mathbb{Z}}_p.$$

Thus, $h(x)=0,x\in G$ and $f\in P_0(G).$□

In Theorem II.1 several sufficient conditions were stated for a group G to have $C_0(G)$ not a ring. Most of these conditions lead to G being split. In Example 3.2 (2) of [11], GAP was used to find a group H, not split, and a function $f\in C_0(H)-P_0(H)$ with $C_0(H)$ not a ring. In the next theorem we give a construction process for a large collection of groups $G\in \text{NAS}(\hspace{0.17em}\text{exp}\hspace{0.17em}p\ne 2)$ to construct a function $f\in C_0(G)$ which shows $C_0(G)$ is not a ring.

First some notation. For $a,b,c,d\in G$ we have a 2 × 2 determinant $$\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=[a,d]-[b,c].$$

For $x,y\in G,x={\sum }_{i=1}^n{x}_i{e}_i+{\sum }_{j=1}^s{u}_j{c}_j,y={\sum }_{i=1}^n{y}_i{e}_i+{\sum }_{j=1}^s{v}_j{c}_j,$ $$[x,y]=\sum _{1\le i<j\le n}(x_i{y}_j-x_j{y}_i)[e_i,e_j]=\sum _{1\le i<j\le n}\left|\begin{array}{cc}{x}_i{e}_i& x_j{e}_i\\ y_i{e}_j& y_j{e}_j\end{array}\right|,$$ similar to the wedge product in multilinear algebra.

Using the above definition of determinant we define a “wedge” product for $x_1,\dots ,x_k,y_1,\dots ,y_k\in G$ by $$\bigwedge \left(\begin{array}{cccc}{x}_1& x_2& \cdots & x_k\\ y_1& y_2& \cdots & y_k\end{array}\right)=\left(\left|\begin{array}{cc}{x}_1& x_2\\ y_1& y_2\end{array}\right|,\left|\begin{array}{cc}{x}_1& x_3\\ y_1& y_3\end{array}\right|,\dots ,\left|\begin{array}{cc}{x}_{k-1}& x_k\\ y_{k-1}& y_k\end{array}\right|\right),$$ a $(\begin{array}{c}k\\ 2\end{array})$-tuple.

For an abelian subgroup, A, of G, $A\supseteq Z(G),$ we choose a basis $\{e_1+Z(G),\dots ,e_{\mathcal{\ell }}+Z(G)\}$ for $A/Z(G)$ and extend this to a basis $\{e_1+Z(G),\dots ,e_{\mathcal{\ell }}+Z(G),e_{\mathcal{\ell }+1}+Z(G),\dots ,e_n+Z(G)\}$ of $G/Z(G).$ Thus without loss of generality we have $G=H+A$ where $H=〈e_{\mathcal{\ell }+1},\dots ,e_n〉$ and $A=〈e_1,\dots ,e_{\mathcal{\ell }}〉+Z(G).$

Theorem

III.3. (Wedge) Let $G=H+A$ as above with $A=Y+Z(G),Y=〈e_1,\dots ,e_{\mathcal{\ell }}〉$. If there exist ${\pi }_{\mathcal{\ell }+1},\dots ,{\pi }_n\in Y$ such that

1. $\bigwedge (\begin{array}{ccc}{e}_{\mathcal{\ell }+1}& \cdots & e_n\\ {\pi }_{\mathcal{\ell }+1}& \cdots & {\pi }_n\end{array})=0,$ and

2. $[e_j,{\pi }_j]\ne 0$ for at least one $j\in \{\mathcal{\ell }+1,\dots ,n\},$

then $C_0(G)$ is not a ring.

Proof.

For suitable ${\pi }_{\mathcal{\ell }+1},\dots ,{\pi }_n\in Y,$ i.e., ${\pi }_i,\mathcal{\ell }+1\le i\le n,$ satisfying i) and ii), we will show that the function $f\in M_0(G),$ defined by $$\begin{array}{c}f\left(\sum _{i=1}^n{x}_i{e}_i\right)=\left[\sum _{i=1}^n{x}_i{e}_i,\sum _{i=\mathcal{\ell }+1}^n{x}_i{\pi }_i\right]\\ =\left[\sum _{i=\mathcal{\ell }+1}^n{x}_i{e}_i,\sum _{i=\mathcal{\ell }+1}^n{x}_i{\pi }_i\right],\mathrm{ }\text{since}\mathrm{ }[e_j,{\pi }_i]=0,1\le j\le \mathcal{\ell },1\le i\le n\end{array}$$ is in $C_0(G),$ but not in $P_0(G).$ Moreover, we’ll show that, for some non-zero $c={\sum }_{i=1}^s{\delta }_i{c}_i\in Z(G),$ $f({\sum }_{i=1}^n{x}_i{e}_i)=f({\sum }_{i=1}^n(-x_i)e_i)=c,$ which shows that $C_0(G)$ contains a non-distributive element, hence $C_0(G)$ is not a ring.

First we show that for arbitrary $x={\sum }_{i=1}^n{x}_i{e}_i,y={\sum }_{i=1}^n{y}_i{e}_i\in G,$ $$f(x)-f(y)=f\left(\sum _{i=1}^n{x}_i{e}_i\right)-f\left(\sum _{i=1}^n{y}_i{e}_i\right)\in \left[\sum _{i=1}^n(x_i-y_i)e_i,G\right]=[x-y,G].$$

Now, for suitable ${\pi }_i,(\mathcal{\ell }+1\le i\le n)$ (those given by i) and ii)): $$\begin{array}{c}f\left(\sum _{i=1}^n{x}_i{e}_i\right)-f\left(\sum _{i=1}^n{y}_i{e}_i\right)=\left[\sum _{i=\mathcal{\ell }+1}^n{x}_i{e}_i,\sum _{i=\mathcal{\ell }+1}^n{x}_i{\pi }_i\right]-\left[\sum _{i=\mathcal{\ell }+1}^n{y}_i{e}_i,\sum _{i=\mathcal{\ell }+1}^n{y}_i{\pi }_i\right]\\ =\sum _{j=\mathcal{\ell }+1}^n\left[x_j{e}_j,\sum _{i=\mathcal{\ell }+1}^n(x_i-y_i){\pi }_i\right]+\sum _{j=\mathcal{\ell }+1}^n\left[(x_j-y_j)e_j,\sum _{i=\mathcal{\ell }+1}^n{y}_i{\pi }_i\right]\\ =L+R,\mathrm{ }\text{where}\mathrm{ }R\in [x-y,G].\end{array}$$

We show that also $L\in [x-y,G]:$ $$\begin{array}{c}L=\sum _{j=\mathcal{\ell }+1}^n\left[x_j{e}_j,\sum _{i=\mathcal{\ell }+1}^n(x_i-y_i){\pi }_i\right]\\ =\sum _{j=\mathcal{\ell }+1}^n\sum _{i=\mathcal{\ell }+1}^n{x}_j(x_i-y_i)[e_j,{\pi }_i]\\ =\sum _{j=\mathcal{\ell }+1}^n\sum _{i=\mathcal{\ell }+1}^n{x}_j(x_i-y_i)[e_i,{\pi }_j],\mathrm{ }\text{by}\text{ i)}\\ =\sum _{j=\mathcal{\ell }+1}^n\sum _{i=\mathcal{\ell }+1}^n\left[(x_i-y_i)e_i,x_j{\pi }_j\right]\\ =\sum _{i=\mathcal{\ell }+1}^n\left[(x_i-y_i)e_i,\sum _{j=\mathcal{\ell }+1}^n{x}_j{\pi }_j\right]\\ \in [x-y,G].\end{array}$$

This shows that $f\in C_0(G).$ By ii), there is an $i_0(\mathcal{\ell }+1\le i_0\le n)$ such that $[e_{i_0},{\pi }_{i_0}]\ne 0.$ So $f(e_{i_0})=[e_{i_0},{\pi }_{i_0}]=[-e_{i_0},-{\pi }_{i_0}]=f(-e_{i_0}).$ It follows that $f°(-id)\ne -(f°id),$ showing that $C_0(G)$ is not a ring.□

Note that condition ii) is necessary here. Otherwise we could have chosen all ${\pi }_i=0$ and i) is still satisfied. But in this case f would be the zero function, hence distributive.

Example

III.4. Let G be given by

Then $G=H+A$ with $H=〈e_1,e_3,e_4〉$ and $A=〈e_2,e_5〉+Z(G).$ We have $\bigwedge (\begin{array}{ccc}{e}_1& e_3& e_4\\ e_5& e_2& e_2\end{array})=0$ and $[e_1,e_5]\ne 0.$ Thus $C_0(G)$ is not a ring.

From Theorem II.1, if G has a maximal abelian normal subgroup A of order $\left|A\right|=p^{n-1+s}$ then $C_0(G)$ is not a ring. As an application of the wedge theorem we consider the case where a maximal abelian normal subgroup, A, has order $p^{n-2+s}$ and G/A is not cyclic. As above we choose a basis $\{e_1+Z(G),\dots ,e_{n-2}+Z(G)\}$ of $A/Z(G)$ and get $G=〈e_{n-1},e_n〉+A,A=〈e_1,\dots ,e_{n-2}〉+Z(G).$

For $i=1,2,\dots ,n-2,[e_i,G]=〈[e_i,e_{n-1}],[e_i,e_n]〉.$ If $[e_i,e_{n-1}]=0$ or $[e_i,e_n]=0,$ then from Theorem II.1, $C_0(G)$ is not a ring. Thus we take $[e_i,e_{n-1}]\ne 0\ne [e_i,e_n]$ for $i\in \{1,2,\dots ,n-2\}.$ Let $A_{n-1}=〈[e_i,e_{n-1}]|1\le i\le n-2〉$ and $A_n=〈[e_i,e_n]|1\le i\le n-2〉.$ If $\{[e_i,e_{n-1}]|1\le i\le n-2\}$ is linearly dependent over ${\mathbb{Z}}_p,$ then ${\sum }_{i=1}^{n-2}{\alpha }_i[e_i,e_{n-1}]=0$ and not all ${\alpha }_i=0.$ So, from $[{\sum }_{i=1}^{n-2}{\alpha }_i{e}_i,e_{n-1}]=0,$ we see that $y={\sum }_{i=1}^{n-2}{\alpha }_i{e}_i$ is in A and $y\ne 0.$ We have $\bigwedge (\begin{array}{cc}{e}_{n-1}& e_n\\ 0& y\end{array})=0$ and $[e_n,y]\ne 0,$ otherwise $y\in Z(G),$ a contradiction. From the Wedge Theorem, $C_0(G)$ is not a ring. Thus we now take $\{[e_i,e_{n-1}]|1\le i\le n-2\}$ to be linearly independent and similarly $\{[e_i,e_n]|1\le i\le n-2\}$ is linearly independent. We have $\left|A_{n-1}\right|=p^{n-2}=\left|A_n\right|.$

Suppose $A_{n-1}\cap A_n\ne \left\{0\right\},$ say $[h,e_{n-1}]=[g,e_n],h,g\in A,$ say $g={\sum }_{i=1}^{n-2}{\beta }_i{e}_i,$ not all ${\beta }_i=0.$ If $[g,e_{n-1}]=0$ then $0={\sum }_{i=1}^{n-2}{\beta }_i[e_i,e_{n-1}],$ a contradiction to the linear independence of $\{[e_i,e_{n-1}]|1\le i\le n-2\}.$ We have $\bigwedge (\begin{array}{cc}{e}_{n-1}& e_n\\ g& h\end{array})=0,h,g\in A$ and $[e_{n-1},g]\ne 0.$ so $C_0(G)$ is not a ring. Consequently $|A_{n-1}+A_n|=p^{2n-4}$ or $C_0(G)$ is a ring.

So we have

Theorem

III.5. Let $G\in \text{NAS}(\hspace{0.17em}\text{exp}\hspace{0.17em}p\ne 2)$ and let $G=H+A,A\in \eta (G)$, A abelian with $Z(G)\subseteq A$ and $|G/A|=p^2$. If $s<2n-4$ then $C_0(G)$ is not a ring.□

We use the notation and definitions from the above discussion. When $s\nless 2n-4,$ then $s\ge 2n-4.$

If $s=2n-4,$ then $[e_{n-1},e_n]=0$ or $[e_{n-1},e_n]\in A_{n-1}+A_n.$ Suppose $[e_{n-1},e_n]\in A_{n-1}+A_n,$ say $[e_{n-1},e_n]={\sum }_{i=1}^{n-2}{\alpha }_i[e_i,e_{n-1}]+{\sum }_{i=1}^{n-2}{\beta }_i[e_i,e_n].$ Let $g={\sum }_{i=1}^{n-2}{\alpha }_i{e}_i$ and $h={\sum }_{i=1}^{n-2}{\beta }_i{e}_i,$ so $[e_{n-1},e_n]=[g,e_{n-1}]+[h,e_n],$ hence $[e_{n-1},e_n+g]=[h,e_n].$ Let ${\widehat{e}}_n=e_n+g$ and note $[e_{n-1},{\widehat{e}}_n]=[h,e_n]=[h,{\widehat{e}}_n]$ so $[e_{n-1}-h,{\widehat{e}}_n]=0$ and $[e_i,e_{n-1}-h]=[e_i,e_{n-1}]$ and $[e_i,e_n+g]=[e_i,e_n],i=1,2,\dots ,n-2.$ By using the basis, $\{e_1,e_2,\dots ,e_{n-2},e_{n-1}-h,e_n+g,c_1,\dots ,c_s\},$ we have $G=〈e_{n-1}-g,e_n+h〉+A$ with $[e_n-g,e_{n-1}+h]=0$ so when $s=2n-4$ we may take $[e_{n-1},e_n]=0.$ When n = 4 we see that G is circular with n even so $C_0(G)$ is not a ring. The case for $n>4,s=2n-4$ remains open.

When $s>2n-4$ and n = 4, then $s=2n-3,$ since $s\le n(n+1)/2=6$ and s = 6 is the full case. For n = 4 and s = 5 one finds via tedious calculations that G is 1-ac. The case n > 4 remains open.

In conclusion, we have identified several further classes of non-abelian p-groups, G, $p\ne 2,$ for which $C_0(G)$ is a ring if and only if G is 1-ac. However, the original conjecture as to whether this is true for all finite non-abelian p-groups, $p\ne 2,$ remains open.

Acknowledgements

Portions of this research were done while the authors were visiting Johannes Kepler Universität, Linz, Austria. They wish to express their appreciation for the gracious hospitality and financial support provided.

Additional information

Funding

The authors also wish to thank the Austrian Science fund (FWF), Project FWF P29931 for financial support.