On Shunkov Groups Saturated with Finite Groups

Abstract

The structure of the group consisting of elements of finite order depends to a large extent on the structure of the finite subgroups of the group under consideration. One of the effective conditions for investigating an infinite group containing elements of finite order is the condition for the group to be saturated with a certain set of groups. The group $G$ is saturated with groups from the set $\mathfrak{X}$ if any finite subgroup of $G$ is contained in the subgroup of $G,$ isomorphic to some group in $\mathfrak{X}.$ The group $G$ is called the group Shunkov, if for any finite subgroup $H$ of $G$ in the factor group $N_G(H)/H$ any two conjugate elements of prime order generate a finite group. If all elements of finite orders in $G$ are contained in a periodic subgroup of $G,$ then it is called the periodic part of $ G $ and is denoted by $T(G).$ It is proved that the Shunkov group of $2$ -range $2$ saturated with finite simple nonabelian groups has a periodic part $T(G)$ isomorphic to one of the groups of the set $\{L_2 (Q), \ A_7, \ L_3 (P), \ U_3 (R), \ M_ {11}, \ U_3 (4) \},$ where $Q,P,R$ is a local finite fields. It is proved that if the Shunkov group $G$ is saturated with finite simple non-Abelian groups, and in any of its finite $2$ -subgroup $K$ all involutions from $K$ lie in the center of $K$, then $G$ has a periodic part $T(G)$ isomorphic to one of the groups of the set $ \{J_1, L_2 (Q), Re (P), U_3 (R), Sz (F) \}$, where $ Q, P, R, F $ are locally finite fields.

Keywords

• the group saturated with the set of groups
• Shunkov group