On periodic groups of $2$-rank one

By the well-known theorems of Burnside, Brauer, and Suzuki \cite[5, Theorem 4.88], if $G$ is a finite group of $2$-rank $1$ with involution $i$, then the involution $iO(G)$ lies in the center of the factor group $G/O(G)$. In $1973$ V. P. Shunkov in the Kourovka Notebook posed Question $4.75$ about the validity of this theorem in the class of periodic groups. The answer is unknown even if the centralizer $C_G(i)$ is quasicyclic, see Question 15.54 of V. D. Mazurov from the Kourovka notebook. The answer to Question 15.54 is affirmative when $G$ acts (sharply) 2-transitively on the set $G/C_G(i)$ (see [2]) and when $C_G(i)$ is a quasicyclic group that is not maximal in $G$.
Recall that an involution $i$ of a group $G$ is called finite if for any $g \in G$ the subgroup $\langle i, i^g \rangle$ is finite. In a periodic group every involution is finite. In the article [MatZam] it was proved that the group $G$ with a malnormal nonmaximal in $G$ $2$-subgroup $C$ and a finite involution $i$ is a locally finite Frobenius group with abelian kernel $[i, G]$ and locally cyclic or (generalized) quaternionic complement $C$. The proof in this work is based on the methods of investigating groups with strongly embedded subgroups and the result of B. Amberg and Ya. P. Sysak [3].
In [4] a partial positive solution to Question 4.75 is obtained under the condition that the involution $i$ of the group $G$ generates a finite subgroup with every element of finite order not divisible by 4. In particular, Question 4.75 is answered positively in the class of conjugate binary finite groups. However, even for the class of conjugate biprimitively finite groups, better known as Shunkov groups, Question 4.75 is still open. The following theorem is announced.

\noindent Theorem. Suppose that a group $G$ satisfies the conditions of the Question 4.75 and any triple of involutions from $G$ generates in $G$ a subgroup different from its commutator subgroup. Then $i O(G) \in Z(G/O(G))$.

In the future we plan to continue investigations of infinite groups of 2-rank 1 and with additional finiteness conditions, including groups saturated with finite Frobenius groups. A group $G$ is saturated with groups from the set of finite groups $\mathfrak {X}$ if every finite subset of elements from $G$ is contained in a subgroup of $G$ which is isomorphic to a group from the set $\mathfrak{X}$. The article [6] based on the results of the study of periodic groups with this condition.

\noindent Acknowledgements. This work is supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement No. 075-02-2021-1388) and by the Russian Foundation for Basic Research according to the project no. 19-01-00566 A.

Список литературы

1. А. И. Созутов, Б. Е. Дураков, “О группах с обособленной $2$-подгруппой”, Матем. заметки, 105:3 (2019), 428–432        ; Math. Notes, 105:3 (2019), 425–428      
2. Н. М. Сучков, “О конечности некоторых точно дважды транзитивных групп”, Алгебра и логика, 40:3 (2001), 344–351      ; Algebra and Logic, 40:3 (2001), 190–193    
3. B. Amberg, Ya. Sysak, “On products of groups with abelian subgroups of small index”, Journal of Group Theory, 2017
4. Б. Е. Дураков, “О некоторых группах 2-ранга один”, Тр. ИММ УрО РАН, 25, № 4, 2019, 64–68      
5. D. Gorenstein, Finite simple groups: an introduction to their classification, Plenum Press, 1982
6. B. E. Durakov, A. I. Sozutov, “On periodic groups saturated with finite Frobenius groups”, Известия Иркутского государственного университета. Серия Математика, 35 (2021), 73–86