Groups of prime-power order having an Abelian centralizer of type (r,1)
Let G be a group of order \(p^ n\), where p is an odd prime, having an element s of order p the centralizer L of which is the direct product of \(\) and a cyclic group \(<t>\) of order \(p^ r\). The aim of the paper is to examine the commutator and power structure of G (assuming \(n>r+2)\). The first assertion is that the class of G is n-r, and there is a unique subgroup of each possible order between L and G. Let \(G=G_ 1>G_ 2>...>G_{n-r}>G_{n-r+1}=1\) be the lower central series of G. Roughly speaking, \(| G_ i:G_{i+1}|\) is in general p for \(i\geq 3\), but there may be bulges in which \(| G_ i:G_{i+1}| =p^ 2.\) The first such bulge will occur with \(i=2h-1\), where \(2\leq h\leq (p+1),\) and there are at most r-2 bulges which occur for \(i=2h-1,\quad 2h-1+p- 1),...,2h-1+r'(p-1).\) Thus the lower central series of G has a tail of factors of order p, and this tail must have length at least p-1. As for the power structure, if \(s_ 1\) is any element of \(G-LG_ 2\) and \(s_ i=[s,s_{i-1}]\) \((i=2,...,n-r)\), then \(s_ i\in G_ i-G_{i+1}\) and \(s^ p_ is_{i+p-1}\in G_{i+p}\) \((i=2,...,n-p-r+1)\). Thus the group generated by the p-th powers of the elements of \(G_ i\) is \(G_{i+p-1}\) for \(i=3,...,n-r-p+2\).
scientific article; zbMATH DE number 3879132