On the commutator subgroups of finite $2$-groups generated by involutions

B. M. Veretennikov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 4, pp. 77-84

Résumé

For a finite group $G$ we denote by $d(G)$ the minimum number of its generators and by $G'$ the commutator group of $G$. In 1975 Ustyuzhaninov published without proof the list of finite $2$-groups generated by three involutions with elementary abelian commutator subgroup. In particular, $d(G') \leq 5$ for such a group $G$. Continuing this research, we pose the problem of classifying all finite $2$-groups generated by $n$ involutions (for any $n\geq 2$) with elementary abelian commutator subgroup. For a finite $2$-group $G$ generated by $n$ involutions with $d(G)=n$, we prove that $$d(G') \leq \left(\begin{array}[c]{c}n\\2 \end{array}\right) + 2 \left(\begin{array}[c]{c}n\\3 \end{array}\right) + \dots + (n-1) \left(\begin{array}[c]{c}n\\n \end{array}\right)$$ for any $n \geq 2$ and that the upper bound is attainable. In addition, we construct for any $n \geq 2$ a finite $2$-group generated by $n$ involutions with elementary abelian commutator subgroup of rank $\left(\begin{array}[c]{c}n\\2 \end{array}\right) + 2 \left(\begin{array}[c]{c}n\\3 \end{array}\right) + \dots + (n-1) \left(\begin{array}[c]{c}n\\n \end{array}\right)$. The method of constructing this group is similar to the method used by the author in a number of papers for the construction of Alperin's finite groups. We obtain $G$ as the consecutive semidirect product of groups of order $2$. We also give an example of an infinite $2$-group generated by involutions with infinite elementary abelian commutator subgroup; the example is obtained from the constructed finite $2$-groups.

Détail

Mots-clés : $2$-group
Keywords: generation by involutions, commutator subgroup.