On groups with a splitting automorphism of prime order
An automorphism \(\varphi\) of a group \(G\) is called a splitting automorphism of prime order \(p\) if \(\varphi^p=1\), \(x\cdot x^\varphi\cdot x^{\varphi^2}\cdot\dots\cdot x^{\varphi^{p-1}}=1\) for all \(x\in G\). For \(\varphi=1\) this is exactly the definition of a group of prime exponent \(p\). If \(G\) is a finite \(p'\)-group then such an automorphism \(\varphi\) is regular. Let \({\mathfrak M}_p\) be the variety of groups with operators, consisting of all groups with a splitting automorphism \(\varphi\) of prime order \(p\). The author [Mat. Sb., Nov. Ser. 130(172), No. 1(5), 120-127 (1986; Zbl 0608.20025)] has proved that the nilpotency class of a \(d\)-generator nilpotent group of \({\mathfrak M}_p\) is bounded by a function depending only on \(d\) and \(p\). Thus the locally nilpotent groups in \({\mathfrak M}_p\) form a subvariety \(LN{\mathfrak M}_p\). Since the variety \({\mathfrak M}_p\) is closely connected with classes of groups of prime exponent \(p\) and groups with regular automorphism of prime order \(p\), the following question arises: Is it true that the subvariety \(LN{\mathfrak M}_p\) is a join of the subvariety \({\mathfrak B}\cap LN{\mathfrak M}_p\) of groups of prime exponent and the subvariety \({\mathfrak N}_{c(p)}\cup{\mathfrak M}_p\) of nilpotent groups of some \(p\)-bounded class? If the answer to the question is affirmative, then the following equalities must hold in any group \(G\) in \(LN{\mathfrak M}_p\): \((\gamma_{c(p)+1}(G))^p=1\), \(\gamma_{c(p)+1}(G^p)=1\). In the present article the author proves that there exist \(p\)-bounded numbers \(k(p)\) and \(l(p)\) such that every group \(G\) in \(LN{\mathfrak M}_p\) satisfies the identities \([x^{p^k(p)}_1,x^{p^k(p)}_2,\dots,x^{p^k(p)}_{h(p)+1}]=1\) which means that the subgroup \(G^{p^{k(p)}}\) is nilpotent of class \(h(p)\); i.e., \(\gamma_{h(p)+1}(G^{p^{k(p)}})=1\) and \([x_1,x_2,\dots,x_{h(p)+1}]^{p^{l(p)}}=1\), where \(h(p)\) is the Higman function bounding the nilpotency class of a nilpotent group with a regular automorphism of prime order \(p\).
scientific article; zbMATH DE number 825568