Finite 2-groups with a self-centralizing elementary Abelian subgroup of order 8. (Q1414034)

M. Suzuki showed that if a nonabelian \(p\)-group \(G\) has a self-centralizing subgroup of order \(p^2\), then \(G\) is of maximal class (the converse, deeper assertion, is also true, and is due to Blackburn). In this fundamental paper the finite \(2\)-groups \(G\) possessing a self-centralizing elementary Abelian subgroup \(E\) of order 8 are classified. This is part of the more general problem of classification of \(p\)-groups containing a self-centralizing Abelian subgroup \(E\) of order \(p^3\) (it is known that if \(E\) is self-centralizing nonabelian of order \(p^3\), then \(G\) is of maximal class; see Proposition 19 in [\textit{Y. Berkovich}, J. Algebra 199, No. 1, 262-280 (1998; Zbl 0917.20015)]). In view of a forthcoming paper of Janko, it remains to consider, for \(p=2\), only the case where \(E\) is cyclic. If \(E\) is normal in \(G\) and \(E\leq\Phi(G)\), then there are exactly six title groups and they are presented in terms of generators and relations (Theorems 2.1 and 2.2). If \(E\) is not normal in \(G\), there are two essentially different possibilities according to \(E<\Phi(G)\), or \(E\nleq\Phi(G)\). It appears that in the first case \(G\) has no normal subgroup \(\cong E\) (Theorem 2.3). Using Theorem 1.1 from [\textit{Z. Janko}, J. Algebra 246, No. 2, 951-961 (2001; Zbl 0992.20012)] the author describes such groups very accurately (Theorem 2.4). The remaining case where \(E\nleq\Phi(G)\), is considered in \S\S 3, 4. It follows from the above results the description of the \(2\)-groups \(G\) containing an involution \(t\) such that \(C_G(t)=\langle t\rangle\times D\), where \(D\) is either dihedral or semidihedral (the case where \(D\) is generalized quaternion is treated in [\textit{Z. Janko}, J. Algebra 245, No. 1, 413-429 (2001; Zbl 0994.20018)]). It follows from the previous results of the author and the results of this paper a description of the \(2\)-groups \(G\) containing an involution \(t\) such that \(| C_G(t)|=8\). It is remarkable that so deep results are achieved by using elementary methods.