Periodic groups saturated by direct products of Suzuki groups and elementary abelian \(2\)-groups
D. Lytkina, A. Duzh
Let \(\mathfrak{M}\) be a nonempty set of finite groups.
A group \(G\) is saturated with groups in \(\mathfrak{M}\)
if every finite subgroup of \(G\) is contained in a subgroup of \(G\) that is isomorphic to an element of \(\mathfrak{M}\), see [3].
A review of the results on the structure of groups saturated by various sets of groups is contained in [1].
In particular, K. Philippov [2] showed that a periodic group saturated by simple groups of Suzuki
is isomorphic to a simple group of Suzuki over a locally finite field of characteristic \(2\).
This paper deals with generalizations of this result. Let
\[\mathfrak{M} = \{Sz (2^{2m+1})\times V_n \mid m = 1, 2, ..., n = 1, 2, ... \},\]
where \(V_n\) is an elementary abelian 2-group of order \(2^{n}\).
Theorem.If \(G\) is a periodic group saturated by \(\mathfrak {M}\)
then \(G \simeq P\times V\), where \(V\) is an elementary abelian \(2\)-group,
and \(P \simeq Sz (Q)\) for some locally finite field \(Q\) of characteristic 2. In particular, \(G\) is locally finite.
References
A. A. Kuznetsov, K. A. Filippov, Groups saturated by specified set of groups, Sib. Elect. Math. Reports, 8 (2011), 230–246.
K. A. Filippov, Groups saturated by finite nonabelian groups and their extensions, Ph.D. Phys.-Math. Thesis, Krasnoyarsk (2006).
À. Ê. Shlepkin, Conjugate biprimitive finite groups that contain nonsolvable subgroups, 3rd Internat. Alg. Conf., Krasnoyarsk (1993), 363.
