On quaternion-free 2-groups
Finite modular groups, that is, those finite groups whose lattice of subgroups is modular, have been classified by \textit{K.~Iwasawa} [J. Fac. Sci. Univ. Tokyo, Sect. I 4, 171-199 (1941; Zbl 0061.02503)] and \textit{F.~Napolitani} [Rend. Sem. Mat. Univ. Padova 39, 296-303 (1967; Zbl 0169.03404)]. A finite \(2\)-group \(P\) is modular if and only if it is \(D_8\)-free, that is, no subgroup of it has a quotient isomorphic to the dihedral group \(D_8\) of order \(8\). In turn, this is equivalent to \(P\) being either both \(D_8\)-free and \(Q_8\)-free, or of the form \(P\cong E\times Q_8\), with \(E\) elementary Abelian. Here \(Q_8\) is the quaternion group of order \(8\). Theorem~1 of the paper under review shows that a modular, quaternion-free finite \(2\)-group has a characteristic Abelian subgroup with a ``good'' metacyclic factor. Theorem~2 gives a neat classification of the finite \(2\)-groups that are \(Q_8\)-free but not \(D_8\)-free.
scientific article; zbMATH DE number 1868097