Large Abelian subgroups of finite \(p\)-groups
Let \(p\) be a prime and let \(G\) be a finite \(p\)-group. Let \(d(G)\) be the maximum of \(| A|\) as \(A\) ranges over the abelian subgroups of \(G\), and \({\mathcal A}(G)\) be the set of all abelian subgroups \(A\) for which \(| A|=d(G)\). Let \(B\) be an abelian subgroup of \(G\) normalized by \(A\) and \(B\) does not normalize \(A\). It is known that there exists \(A^*\in{\mathcal A}(G)\) such that \(A^*\leq AB\), \(A\cap B<A^*\cap B\) and \(A^*\) normalizes \(A\). The author extends this result by allowing \(B\) to have nilpotence class 2 instead of necessarily being abelian when \(p\) is odd. In particular it is proved, that if \(G\) is a finite \(p\)-group, \(p\) is an odd prime, \(A\in{\mathcal A}(G)\), \(B\) is a subgroup of \(G\) that does not normalize \(A\), \(B\) has nilpotence class at most 2 and is normalized by \(A\) or, if \(p\geq 5\), \(A\triangleleft\langle A^B\rangle\), and \([A,u;3]=1\) for every \(u\in B\), then there exists \(A^*\in{\mathcal A}(G)\) such that \(A^*\leq\langle A^B\rangle\) and \(A^*\) normalizes \(A\) and if \(A\) normalizes \(B\) then \(| A^*\cap B|\geq| A\cap B|\).
scientific article; zbMATH DE number 1089125
scientific article