B.M. Veretennikov. On the commutator subgroups of finite 2-groups generated by involutions ... P. 77-84

For a finite group $G$ we denote by $d(G)$ the minimum number of its generators and by $G'$ the commutator group of $G$. 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 the first section we establish the inequality for $d(G')$, and in the second section 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 $G$ is similar to the method used by the author in a number of papers for the construction of Alperin's finite groups. Using the known theorem on cyclic extensions, we obtain $G$ as the consecutive semidirect product of groups of order 2. In the end of the paper, we give an example of an infinite 2-group generated by involutions with infinite elementary abelian commutator; the example is obtained from the constructed finite 2-groups.

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

The paper was received by the Editorial Office on April 10, 2017.

Boris Mihajlovich Veretennikov, Cand. Sci. (Phys.-Math.), Ural Federal University, Yekaterinburg, 620002 Russia,
e-mail: boris@veretennikov.ru

REFERENCES

1.   Ustyuzhaninov A.D. Finite 2-groups generated by exactly three involutions. All-union algebr. symposium (1975), Abstracts, part I, Gomel, 1975, p. 72 (in Russian).

2.   Kargapolov M.I., Merzljakov J.I. Fundamentals of the Theory of Groups. New York: Springer-Verlag, 1979, 203 p. ISBN: 978-1-4612-9966-0 . Original Russian text published in Kargapolov M.I., Merzlyakov Yu.I. Osnovy teorii grupp. Moscow, Nauka Publ, 1977, 240 p.

3.   Veretennikov B.M. Finite Alperin 2-groups with cyclic second commutants. Algebra Logic, 2011, vol. 50, pp. 226–244. doi: 10.1007/s10469-011-9137-6 .