Large Abelian subgroups of groups of prime exponent (Q5936197)

Let \(G\) be a finite \(p\)-group. Suppose \(G\) has an elementary Abelian subgroup of order \(p^n\). One might ask whether such a \(G\) has a normal elementary Abelian subgroup of order \(p^n\). The dihedral group of order 16 with \(n=2\) shows that, in general, the answer is no. J.~L.~Alperin has constructed counterexamples for each prime \(p\). There are, however, situations when the answer is yes, such as when \(p\) is odd and greater than \(4n-7\), or when \(p\leq 3\) and \(G\) has exponent \(p\); for these and other results of this kind see the paper of \textit{J. L. Alperin} and \textit{G. Glauberman} [J. Algebra 203, No. 2, 533-566 (1998; Zbl 0964.20011)].NEWLINENEWLINENEWLINEThe goal of the paper under review is to construct a series of counterexamples \(G\) of exponent \(p\), with \(p\geq 5\). Each \(G\) is constructed as a semi-direct product of a group \(R\) of nilpotency class \(2\) by a cyclic group generated by an automorphism of order \(p\). The Lie method is applied in an ingenious way. First the group \(R\) is built up starting from a Lie algebra, and \(\alpha\) is defined as the exponential of a nilpotent linear transformation \(\delta\) of the related Lie algebra. To check that the resulting group \(G\) has exponent \(p\), one might verify certain properties of \(\alpha\). Since this would be rather involved, these properties are conveniently translated instead into properties of \(\delta\).