Mathematical logic, algebra and number theory

On finite groups generated by involutions

B. M. Veretennikov

Ural Federal University, 19 Mira street, 620002 Ekaterinburg, Russia

Abstract: All groups in the abstract are finite. In theorem $1$ we prove that any group $A$, generated by $n$ involutions ($n \geq 3$), is a section $G/N$ of some group $B$, generated by three involutions (respectively, generated by an element of order $n$ and involution) in which $B/G$ is isomorphic $D_{2n}$ (respectively, $Z_n$). In theorem $2$ we consider the case when $A$ is a $2$-group. In theorem 3 and 4 we prove that any $2$-group is a section of a $2$-group generated by $3$ involutions and a section of a $2$-group generated by element of order $2^m$ and involution ($m$ may be arbitrary integer more than $1$). In the last part of the paper we construct some examples of $2$-groups, generated by $3$ involutions and of $2$-groups, generated by an element and involution of derived lengths $4$ and $3$ respectively.

Keywords: finite group generated by involutions; finite group generated by three involutions, finite $2$-group, Alperin group, definition of group by means of generators and defining relations.

UDC: 512.54

MSC: 20B05

Received February 1, 2016, published May 24, 2016

DOI: 10.17377/semi.2016.13.037

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