Non-abelian tensor square and related constructions of $p$-groups

TL;DR

This paper investigates the structure of non-abelian tensor squares and related constructions in finite potent and powerful p-groups, establishing embedding properties and exponent bounds.

Contribution

It introduces new embedding results and exponent divisibility properties for non-abelian tensor squares of potent p-groups.

Findings

Potent embedding of [G,G^φ] in ν(G) for finite potent p-groups

Exponent of ν(G) divides p times the exponent of G for potent p-groups

Analysis of weak commutativity in powerful p-groups

Abstract

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $[G,G^{\varphi}]$ by $G \times G$. We prove that if $G$ is a finite potent $p$-group, then $[G,G^{\varphi}]$ and the $k$-th term of the lower central series $\gamma_k(\nu(G))$ are potently embedded in $\nu(G)$ (Theorem A). Moreover, we show that if $G$ is a potent $p$-group, then the exponent $\exp(\nu(G))$ divides $p \cdot \exp(G)$ (Theorem B). We also study the weak commutativity construction of powerful $p$-groups (Theorem C).

Equations

$$g{g}_1\otimes h=(g^{g_1}\otimes h^{g_1})(g_1\otimes h)\text{mbox}andg\otimes h{h}_1=(g\otimes h_1)(g^{h_1}\otimes h^{h_1})$$

$${\pi }_3(SK(G,1))\cong \mu (G),$$

$$1\to \Delta (G)\to \mu (G)\to H_2(G)\to 1,$$

$$\nu (G):=⟨G\cup G^{\phi }|[g_1,{g_2}^{\phi }{]}^{g_3}=[{g_1}^{g_3},({g_2}^{g_3}{)}^{\phi }]=[g_1,{g_2}^{\phi }{]}^{{g_3}^{\phi }},g_i\in G⟩.$$

$$\chi (G)=⟨G\cup G^{\phi }\mid [g,g^{\phi }]=1,\forall g\in G⟩.$$

$$\frac{\nu (G)}{\Delta (G)}\cong \frac{\chi (G)}{R(G)}.$$

$$(xy{)}^{p^k}\equiv x^{p^k}{y}^{p^k}(\text{ mod }{\gamma }_2(L{)}^{p^k}{\gamma }_p(L{)}^{p^{k-1}}{\gamma }_{p^2}(L{)}^{p^{k-2}}{\gamma }_{p^3}(L{)}^{p^{k-3}}\cdots {\gamma }_{p^k}(L)),$$

$$[x,y{]}^{p^k}\equiv [x^{p^k},y](\text{ mod }{\gamma }_2(M{)}^{p^k}{\gamma }_p(M{)}^{p^{k-1}}{\gamma }_{p^2}(M{)}^{p^{k-2}}\dots {\gamma }_{p^k}(M)),$$

$$\frac{[G,G^{\phi }]}{\mu (G)}\cong G^{\prime }.$$

$$[G^p,G]\equiv [G,G{]}^p(\text{mod}{\gamma }_{p+1}(G)).$$

$${\gamma }_{n+1}(\nu (G))={\gamma }_{n+1}(G){\gamma }_{n+1}(G^{\phi })[{\gamma }_n(G),G^{\phi }].$$

$$[x^p,y^{\phi }]\equiv [x,y^{\phi }{]}^p(\text{mod}{\gamma }_2(M{)}^p{\gamma }_p(M)),$$

$$\begin{array}[]{lcl}[\gamma_{k+1}(\nu(G)),_{p-2}\nu(G)]&\leqslant&[(\gamma_{k}(\nu(G)))^{p},\nu(G)]\\ &\leqslant&[\gamma_{k}(\nu(G)),\nu(G)]^{p}[\nu(G),\ _{p}\ \gamma_{k}(\nu(G))]\\ &\leqslant&(\gamma_{k+1}(\nu(G))^{p}[\gamma_{k+1}(\nu(G)),\ _{p-2}\ \nu(G),\nu(G)].\end{array}$$

$$[{\gamma }_{k+1}(\nu (G)){,}_{p-2}\nu (G)]⩽({\gamma }_{k+1}(\nu (G){)}^p,$$

$${\gamma }_3(\chi (G))⩽[\chi (G),\chi (G){]}^p.$$

$${\gamma }_3(\chi (G))⩽[G^{\prime },G^{\phi }]⩽[G^p,G^{\phi }].$$

$$[x^p,y^{\phi }]\equiv [x,y^{\phi }{]}^p(\text{mod}{\gamma }_2(M{)}^p{\gamma }_p(M)),$$

$$D(G)=⟨[a,b^{\phi }],[a,c^{\phi }],[b,c^{\phi }],[a,b^{\phi },c]⟩\cong {\mathbb{Z}}_2^4$$

$$[D(G),G]=⟨[a,b^{\phi },c]⟩\cong {\mathbb{Z}}_2.$$

$${\gamma }_{p-1}(\chi (G))⩽[\chi (G{)}^{\prime },\chi (G)]⩽\chi (G{)}^p.$$

Full text

1. Introduction

The non-abelian tensor square $G\otimes G$ of a group $G$, as introduced by R. Brown and J. L. Loday [2, 3], is defined to be the group generated by all symbols $\;\,g\otimes h,\;g,h\in G$, subject to the relations

[TABLE]

for all $g,g_{1},h,h_{1}\in G$, where we write $x^{y}$ for the conjugate $y^{-1}xy$ of $x$ by $y$, for any elements $x,y\in G$. In the same paper, R. Brown and J. L. Loday show that the third homotopy group of the suspension of an Eilenberg-MacLane space $K(G,1)$ satisfies

[TABLE]

where $\mu(G)$ is the kernel of the derived map $G\otimes G\to G^{\prime}$, given by $g\otimes h\mapsto[g,h]$. Consider the following short exact sequence, as in [3],

[TABLE]

where $\Delta(G)=\langle g\otimes g\mid g\in G\rangle$ and $H_{2}(G)$ is the second homology group of the group $G$. It has been shown in [10] that $H_{2}(G)\cong M(G)$, where $M(G)$ is the Schur multiplier of $G$. See also [16, Chapter 2 and 3].

The study of the non-abelian tensor square of groups from a group theoretic point of view was initiated by R. Brown, D. L. Johnson and E. F. Robertson [1]. We observe that the defining relations of the non-abelian tensor square can be viewed as abstractions of commutator relations; thus in [15], N. R. Rocco considered the following construction. Let $G$ be a group and let $\varphi:G\rightarrow G^{\varphi}$ be an isomorphism ($G^{\varphi}$ is a copy of $G$, where $g\mapsto g^{\varphi}$, for all $g\in G$). Define the group $\nu(G)$ to be

[TABLE]

The motivation for studying $\nu(G)$ is the commutator connection: indeed, the map $\Phi:G\otimes G\rightarrow[G,G^{\varphi}]$, defined by $g\otimes h\mapsto[g,h^{\varphi}]$, for all $g,h\in G$, is an isomorphism [15, Proposition 2.6]. Therefore, from now on we shall identify the non-abelian tensor square $G\otimes G$ with the subgroup $[G,G^{\varphi}]$ of $\nu(G)$ and write $[g,h^{\varphi}]$ instead of $g\otimes h$, for all $g,h\in G.$ For a fuller treatment we refer the reader to [8, 12].

Our purpose is to study the structure of the non-abelian tensor square and related constructions of finite powerful and potent $p$-groups.

Let $p$ be a prime number. A finite $p$-group $G$ is said to be powerful if $p>2$ and $G^{\prime}\leqslant G^{p}$, or $p=2$ and $G^{\prime}\leqslant G^{4}$. We can define a more general class of $p$-groups. We call a finite $p$-group potent if $p>2$ and $\gamma_{p-1}(G)\leqslant G^{p}$, or $p=2$ and $G^{\prime}\leqslant G^{4}$. Note that the family of potent $p$-groups contains all powerful $p$-groups. Recall that a subgroup $N$ of $G$ is potently embedded in $G$ if $[N,G]\leqslant N^{4}$, for $p=2$, or $[N,_{p-2}G]\leqslant N^{p}$ for odd prime ($N$ is powerfully embedded in $G$ if $[N,G]\leqslant N^{4}$, for $p=2$, or $[N,G]\leqslant N^{p}$ for odd prime). More information on finite powerful and potent $p$-groups can be found in [4] and [6], respectively.

In [11], Moravec proved that if $G$ is a powerful $p$-group, then the non-abelian tensor square $[G,G^{\varphi}]$ and the derived subgroup $\nu(G)^{\prime}$ are powerfully embedded in $\nu(G)$. Moreover, the exponent $\exp(\nu(G)^{\prime})$ divides $\exp(G)$. We extend these results to potent $p$-groups.

Theorem A**.**

Let $p$ be a prime and $G$ a finite potent $p$-group.

• (a)

The non-abelian tensor square $[G,G^{\varphi}]$ is potently embedded in $\nu(G)$;

• (b)

If $k\geqslant 2$, then the $k$-th term of the lower central series $\gamma_{k}(\nu(G))$ is potently embedded in $\nu(G)$.

Theorem B**.**

Let $p$ be a prime and $G$ a $p$-group with $\exp(G)=p^{e}$.

• (a)

If $G$ is potent, then $\exp(\nu(G))$ divides $p^{e+1}$;

• (b)

If $\gamma_{p-2}(G)\leq G^{p}$, then $\nu(G)$ is a potent $p$-group. In particular, $\exp(\nu(G))=p^{e}$.

The following results are immediate consequences of Theorem B.

Corollary 1.1**.**

Let $p\geqslant 5$ be a prime and $G$ a finite powerful $p$-group. Then $\nu(G)$ is a potent $p$-group. In particular, $\exp(\nu(G))=\exp(G)$.

Corollary 1.2**.**

Let $p$ be a prime and $G$ a finite potent $p$-group. Then the $\exp(M(G))$ and $\exp(\mu(G))$ divide $p\cdot\exp(G)$.

Now, we study the weak commutativity construction of powerful $p$-groups. As before, $G^{\varphi}$ denotes an isomorphic copy of $G$ via $\varphi:G\rightarrow G^{\varphi}$, $g\mapsto g^{\varphi}$, for all $g\in G$. The following group construction was introduced and studied in [17]

[TABLE]

The weak commutativity group $\chi(G)$ maps onto $G$ by $g\mapsto g$, $g^{\varphi}\mapsto g$ with kernel $L(G)=\left\langle g^{-1}g^{\varphi}\mid g\;\in G\right\rangle$, and it maps onto $G\times G$ by $g\mapsto\left(g,1\right),g^{\varphi}\mapsto\left(1,g\right)$ with kernel $D(G)=[G,G^{\varphi}]$. It is an important fact that $L(G)$ and $D(G)$ commute. Define $T(G)$ to be the subgroup of $G\times G\times G$ generated by $\{(g,g,1),(1,g,g)\mid g\in G\}$. Then $\chi(G)$ maps onto $T(G)$ by $g\mapsto\left(g,g,1\right)$, $g^{\varphi}\mapsto\left(1,g,g\right)$, with kernel $W(G)=L(G)\cap D(G)$, an abelian group. In particular, the quotient $\chi(G)/W(G)$ is isomorphic to a subgroup of $G\times G\times G$. A further normal subgroup of $\chi(G)$ is $R(G)=[G,L(G),G^{\varphi}]$, where the quotient $W(G)/R(G)$ is isomorphic to the Schur multiplier $M(G)$. Moreover, in [15, 16], it was proved that the constructions $\chi(G)$ and $\nu(G)$ have large isomorphic quotients. More precisely,

[TABLE]

See [15, Remark 2] and [16, Remark 4]) for more details. The group $\chi(G)$ inherits many properties of the argument $G$; for instance, if $G$ is a finite $\pi$-group ($\pi$ a set of primes), nilpotent, solvable, locally nilpotent, or polycyclic-by-finite group, then so is $\chi(G)$ [17, 14, 7, 9].

The following result is an extension of Moravec’s results [11] in the context of the weak commutativity construction.

Theorem C**.**

Let $p$ be an odd prime and $G$ a powerful $p$-group. Then the subgroups $\chi(G)^{\prime}$ and $D(G)$ are powerfully embedded in $\chi(G)$.

The above theorem is no longer valid if we drop the assumption that $p$ is an odd prime (see Remark 4.1, below).

The paper is organized as follows. In Section 2 we summarize without proofs some results on finite $p$-groups. In the third section we prove Theorems A and B. The proof of Theorem C is given in Section 4.

2. Finite $p$-groups

Let $G$ be a $p$-group. Consider the following subgroups of $G$: $\Omega_{n}(G)=\langle g\in G\ |\ g^{p^{n}}=1\rangle$ and $G^{p^{n}}=\langle g^{p^{n}}\ |\ g\in G\rangle$. If $G$ is abelian, we have the following simple description of these subgroups:

1. (1)

$\Omega_{n}(G)=\{g\in G\ |\ g^{p^{n}}=1\}$. 2. (2)

$G^{p^{n}}=\{g^{p^{n}}\ |\ g\in G\}$.

Moreover, we have that:

1. (3)

$|G^{p^{n}}|=|G:\Omega_{n}(G)|$ for all $n$.

A finite $p$-group $G$ is called power abelian if it satisfies these three conditions for all $n$. It is a very interesting problem to determine which groups are power abelian. Recently, it was proved in [6] that potent $p$-groups are also power abelian, for odd primes.

The next result are basic facts about finite $p$-groups (see for instance [6]).

Lemma 2.1**.**

Let $G$ be a finite $p$-group and $N$, $M$ normal subgroups of $G$. If $N\leq M[N,G]N^{p}$ then $N\leq M$

The following theorem is known as P. Hall’s collection formula.

Theorem 2.2**.**

Let $G$ be a $p$-group and $x,y$ elements of $G$. Then for any $k\geq 0$ we have

[TABLE]

where $L=\langle x,y\rangle$. We also have that

[TABLE]

where $M=\langle x,[x,y]\rangle$.

The next result is a consequence of P. Hall’s formula.

Lemma 2.3**.**

Let $G$ be a finite $p$-group and $N$, $M$ normal subgroups of $G$. Then $[N^{p},M]\leq[N,M]^{p}[M,_{p}N]$.

In [6, Theorem 2.1], they prove the following useful lemma.

Lemma 2.4**.**

If $G$ is a potent $p$-group, then $\gamma_{k+1}(G)\leqslant\gamma_{k}(G)^{4}$, for $p=2$, and $\gamma_{p-1+k}(G)\leqslant(\gamma_{k+1}(G))^{p}$, for $p\geq 3$.

3. Proofs of Theorems A and B

For the convenience of the reader we repeat some relevant definitions in the context of the non-abelian tensor square (cf. [16]). Recall that there is an epimorphism $\rho:\nu(G)\to G$, given by $g\mapsto g$, $h^{\varphi}\mapsto h$, which induces the derived map $\rho^{\prime}:[G,G^{\varphi}]\to G^{\prime}$, $[g,h^{\varphi}]\mapsto[g,h]$, for all $g,h\in G$. In the notation of [16, Section 2], let $\mu(G)$ denote the kernel of $\rho^{\prime}$, a central subgroup of $\nu(G)$. In particular,

[TABLE]

The next proposition will be used in the proofs of Theorems A and B.

Proposition 3.1**.**

Let $p\geq 3$ be a prime and $n\in\mathbb{N}$ such that $1<n<p$. Suppose that $G$ is a finite $p$-group such that $\gamma_{n}(G)\leqslant G^{p}$. Then $\gamma_{n+1}(\nu(G))\leqslant\gamma_{2}(\nu(G))^{p}$.

Proof.

Clearly, we can assume that $(\gamma_{2}(\nu(G)))^{p}=[\nu(G),\nu(G)]^{p}=1$. In particular, we have that $[G,G]^{p}=[G,G^{\varphi}]^{p}=1$.

As $\gamma_{n}(G)\leqslant G^{p}$, we obtain that $\gamma_{n+1}(G)\leqslant[G^{p},G]$. On the other hand, by the Phillip Hall’s formula we know that

[TABLE]

Now, by Lemma 2.4 $\gamma_{p+1}(G)\leqslant\gamma_{2}(\nu(G))^{p}$. Therefore $[G^{p},G]\leqslant[G,G]^{p}=1$. As a consequence the nilpotency class of $G$ and $G^{\varphi}$ is at most $n$. Recall that $\gamma_{n+2}(\nu(G))=\gamma_{n+2}(G)\gamma_{n+2}(G^{\varphi})[\gamma_{n+1}(G),G^{\varphi}]$ by [15, Theorem 3.1]. Thus, we obtain that the nilpotency class of $\nu(G)$ is at most $n+1$.

We will show that in fact $\gamma_{n+1}(\nu(G))=1$. Again, by [15, Theorem 3.1],

[TABLE]

Then $\gamma_{n+1}(\nu(G))=[\gamma_{n}(G),G^{\varphi}]\leqslant[G^{p},G^{\varphi}]$. It is sufficient to prove that each generator of $[G^{p},G^{\varphi}]$ is trivial. Let $x\in G$ and $y^{\varphi}\in G^{\varphi}$. Using the Phillip Hall’s formula we have that

[TABLE]

where $M=\langle x,[x,y^{\varphi}]\rangle$.

Note that $\gamma_{p}(M)\leqslant\gamma_{p+1}(\nu(G))$. Hence $\gamma_{2}(M)^{p}\leqslant\gamma_{2}(\nu(G))^{p}=1$ and $\gamma_{p}(M)\leqslant\gamma_{p+1}(\nu(G))\leqslant\gamma_{n+2}(\nu(G))=1$, since $p+1\geq n+2$.

Therefore $[x^{p},y^{\varphi}]=1$ for any $x,y\in G$ and so $[G^{p},G^{\varphi}]=1$ as desired. ∎

We are now in a position to prove Theorem A.

Proof.

(a). Recall that the definitions of powerful and potent $p$-groups coincide for $p=2$ and $3$. Using Moravec’s result [11], the non-abelian tensor square $[G,G^{\varphi}]$ is powerful embedded in $\nu(G)$, when $G$ is a powerful $p$-group. Now, it remains to consider potent $p$-groups with $p\geq 5$.

Hence we need to prove that $[[G,G^{\varphi}],_{p-2}\nu(G)]\leqslant[G,G^{\varphi}]^{p}$. Suppose that $[G,G^{\varphi}]^{p}=1$. Consider the derived map $\rho^{\prime}:[G,G^{\varphi}]\to G^{\prime}$, given by $[g,h^{\varphi}]\mapsto[g,h]$. Since $[G,G^{\varphi}]^{p}=1$ and $Im(\rho^{\prime})=G^{\prime}$, we deduce that $(G^{\prime})^{p}=1$. By Lemma 2.4, $\gamma_{p}(G)=1=\gamma_{p}(G^{\varphi})$.

Observe that $[[G,G^{\varphi}],_{p-2}\nu(G)]\leqslant\gamma_{p}(\nu(G))$ and, by [15, Theorem 3.1], $\gamma_{p}(\nu(G))=\gamma_{p}(G)\gamma_{p}(G^{\varphi})[\gamma_{p-1}(G),G^{\varphi}]$. Therefore $\gamma_{p}(\nu(G))=[\gamma_{p-1}(G),G^{\varphi}]\leqslant[G^{p},G^{\varphi}]$.

Let $x\in G$ and $y^{\varphi}\in G^{\varphi}$. Using the Phillip Hall’s formula we have that

[TABLE]

where $M=\langle x,[x,y^{\varphi}]\rangle$. Note that $\gamma_{p}(M)\leqslant\gamma_{p+1}(\nu(G))$. Hence $\gamma_{2}(M)^{p}\leqslant\gamma_{2}(\nu(G))^{p}=1$ and $\gamma_{p}(M)\leqslant\gamma_{p+1}(\nu(G))=1$. Consequently, $[x^{p},y^{\varphi}]=1$ for any $x,y\in G$ and so $[G^{p},G^{\varphi}]=1$. Therefore $[[G,G^{\varphi}],_{p-2}\nu(G)]=1$, as wished.

(b). We will prove by induction on $k$. For $k=2$ apply Proposition 3.1 for $n=p-1$. Suppose by induction hypothesis that $[\gamma_{k}(\nu(G)),_{p-2}\nu(G)]\leqslant(\gamma_{k}(\nu(G)))^{p}$. By Lemma 2.3,

[TABLE]

Therefore, by Lemma 2.1 we have

[TABLE]

which is the desired conclusion. ∎

Proof of Theorem B.

(a). In [5] it was proved that if $P$ is a finite $p$-group such that $\gamma_{k(p-1)}(P)\leq\gamma_{r}(P)^{p^{s}}$ for some $r$ and $s$ such that $k(p-1)<r+s(p-1)$, then the exponent of $\Omega_{i}(P)$ is at most $p^{i+k-1}$ for all $i$.

Note that $\gamma_{2(p-1)}(\nu(G))=[\gamma_{p}(\nu(G)),_{p-2}\nu(G)]$. Then using Theorem A (b) we conclude that $\gamma_{2(p-1)}(\nu(G))\leq\gamma_{p}(\nu(G))^{p}$. Now, by [5] we have that $\exp(\Omega_{e}(\nu(G)))=p^{e+1}$. In particular, $\exp(\nu(G))=p^{e+1}$ since $\nu(G)$ is generated by $G$ and $G^{\varphi}$.

(b). Since $\gamma_{p-2}(G)\leqslant G^{p}$, by Proposition 3.1, we conclude that $\gamma_{p-1}(\nu(G))\leq[\nu(G),\nu(G)]^{p}$ and so $\nu(G)$ is a potent $p$-group. By [6] potent $p$-groups are power abelian $p$-groups, whenever $p$ is an odd prime. As $\nu(G)$ is generated by $G$ and $G^{\varphi}$ we have $\exp(\nu(G))=\exp(G)$. The proof is complete. ∎

4. Proof of Theorem C

For the convenience of the reader we repeat some relevant definitions in the context of the weak commutativity construction (cf. [17]). Let $G$ be a group. The weak commutativity construction of $G$ is defined as $\chi(G)=\langle G\cup G^{\varphi}\mid[g,g^{\varphi}]=1,\forall g\in G\rangle$. Consider the subgroup $D(G)=[G,G^{\varphi}]$ of $\chi(G)$.

Theorem C**.**

Let $p$ be an odd prime and $G$ a powerful $p$-group. Then the subgroups $\chi(G)^{\prime},\;D(G)$ are powerfully embedded in $\chi(G)$.

Proof.

We need to prove that

[TABLE]

Clearly, we can assume that $(\gamma_{2}(\chi(G)))^{p}=[\chi(G),\chi(G)]^{p}=1$. Moreover, by Lemma 2.1, we can suppose that $(\gamma_{2}(\chi(G)))^{p}\gamma_{4}(\chi(G))=1$. In particular, we have that $[G^{\varphi},G^{\varphi}]^{p}=[G,G]^{p}=[G,G^{\varphi}]^{p}=1$. Since $G$ is powerful, it follows that $G$ and $G^{\varphi}$ are nilpotent groups of class at most 2. By [14, Lemma 3.2.1], $\gamma_{3}(\chi(G))\leqslant[G^{\prime},G^{\varphi}].$ Since $G$ is a powerful $p$-group, we deduce that

[TABLE]

Let $x\in G$ and $y^{\varphi}\in G^{\varphi}$. Using the Phillip Hall’s formula we have that

[TABLE]

where $M=\langle x,[x,y^{\varphi}]\rangle$. Note that $\gamma_{p}(M)\leqslant\gamma_{p+1}(\chi(G))$. Hence $\gamma_{2}(M)^{p}\leqslant\gamma_{2}(\chi(G))^{p}=1$ and $\gamma_{p}(M)\leqslant\gamma_{p+1}(\chi(G))\leqslant\gamma_{4}(\chi(G))=1$.

Consequently, $[x^{p},y^{\varphi}]=1$ for any $x,y\in G$ and so $[G^{p},G^{\varphi}]=1$, this establishes that $\chi(G)^{\prime}$ is powerfully embedded in $\chi(G)$.

It remains to prove that $D(G)$ is powerfully embedded in $\chi(G)$, that is, $[D(G),\chi(G)]\leqslant D(G)^{p}.$ Assuming that $D(G)^{p}=1$, by [17, Proposition 4.1.4] we get that $[G,G]^{p}=1$, which follows as above. ∎

Remark 4.1**.**

Theorem C does not hold assuming $p$=2. Taking $G=\left<a,b,c\right>$ being an elementary abelian 2-group of rank 3, we have that $\chi(G)$ is a nilpotent group of class 3, where

[TABLE]

and

[TABLE]

For $p\geqslant 5$ we obtain the following bound to the exponent $\exp(\chi(G))$, when $G$ is a powerful group.

Corollary 4.2**.**

Let $G$ be finite powerful $p$-group with $p\geqslant 5$. Then $\chi(G)$ is potent. Moreover, the exponent $\exp(\chi(G))$ divides $\exp(G)$.

Proof.

By Theorem C, $[\chi(G)^{\prime},\chi(G)]\leqslant\chi(G)^{p}$. In particular,

[TABLE]

As $p\geqslant 5$, we have $G$ is a potent $p$-group and so, the exponent $\exp(\chi(G))=\exp(G)$, which completes the proof. ∎

Acknowledgment

The authors wish to thank professor Noraí Romeu Rocco for interesting discussions. This work was partially supported by FAPDF - Brazil, Grant: 0193.001344/2016.