On Shunkov groups saturated with finite simple groups
K. Philippov and A. Philippova
A group \(G\) is saturated with groups in a set of groups
\(\mathfrak M\) if every finite subgroup of \(G\) is contained in some group in \(\mathfrak M\).
In [2], it is proved that if a periodic group \(G\) is saturated with finite simple nonabelian groups and, for every finite 2-subgroup \(K\) of \(G\), all involutions of \(K\) lie in the center of \(K\) then
\(G\) is isomorphic to one of the following groups: \(J_1\), \(L_2(Q)\), \(Re(Q)\), \(U_3(Q)\), \(Sz(Q)\) for a suitable
locally finite field \(Q\).
In this work, we prove
Theorem. Let a Shunkov group \(G\) be saturated with finite simple nonabelian groups and, for every finite 2-subgroup \(K\) of \(G\), let all involutions of \(K\) lie in the center of \(K\). Then \(G\) possesses periodic part
\(T(G)\) which is isomorphic to one of the following groups: \(J_1\), \(L_2(Q)\), \(Re(Q)\), \(U_3(Q)\), \(Sz(Q)\) for a suitable locally finite field \(Q\).
References
A. K. Shlyopkin, Conjugate biprimitive finite groups that contain nonsolvable subgroups, 3rd Internat. Conf. on Algebra (August 23–28, 1993), Abst., Krasnoyarsk.
K. A. Philippov, On periodic groups that are saturated with finite simple groups,
Sib. Math. J., 53, N2 (2012), 430–438.
