On weak commutativity in \(p\)-groups (Q6060818)

Let \(G\) be a group and let \(G^{\varphi}\) be a copy of the group \(G\), where \(\varphi: G \rightarrow G^{\varphi}\) is the isomorphism given by \(g \mapsto g^{\varphi}\). \textit{S. Sidki} [J. Algebra 63, 186--225 (1980; Zbl 0442.20014)] introduced and analyzed the weak commutativity group \[ \chi(G)=\big \langle (g, g^{\varphi}) \mid g\in G, g^{\varphi} \in G^\varphi, [g,g^{\varphi}]=1 \big \rangle. \] In [loc. cit.], Sidki obtains some bounds for the exponent \(\exp(G)\) for a finite group \(G\). These bounds depend on the exponents of the group and on the Schur multiplier of some sections of \(G\). In the paper under review, the authors present new bounds for the exponent of \(G\), when \(G\) is a finite \(p\)-group. Among other things, they prove the following two results (where \(D(G)=[G,G^{\varphi}]=\langle [g,h^{\varphi}] \mid g,h \in G \rangle\)). Theorem 1.2: Let \(p\) be an odd prime and \(G\) a \(p\)-group of nilpotency class at most \(2p-3\). Then \(\exp(D(G))\) divides \(p\cdot \exp(G)\) and \(\exp(\chi(G))\) divides \(p \cdot \exp(G)^{2}\). Theorem 1.3: Let \(p\) be an odd prime and \(G\) a \(p\)-group of nilpotency class \(c\). Then \(\exp(D(G))\) divides \(\exp(G)^{n+1}\) and \(\exp(\chi(G))\) divides \(\exp(G)^{n+2}\), where \(n=\lceil \log_{p-1} (c+1)-\log_{p-1}(p)\rceil\). The authors prove similar results in the case where \(p=2\).