The periodic groups saturated by finitely many finite simple groups

Abstract

Denote by M the set whose elements are the simple 3-dimensional unitary groups U 3 (q) and the linear groups L 3 (q) over finite fields. We prove that every periodic group, saturated by the groups of a finite subset of M, is finite.Keywords: saturation of a group by a set of groups, periodic group Introduction. A group G is called saturated by groups in some set R if each finite subgroup of G is contained in a subgroup of G isomorphic to some group in R. Take some group G saturated by groups in R and suppose that to X ∈ R there is a subgroup L of G isomorphic to X. In this case we say that G is saturated by R and call R a saturating group set for G.It was conjectured [1] that each periodic group is finite that is saturated by finitely many nonabelian simple groups. This conjecture was confirmed in the case that every element of a saturating group set R is a finite simple group the centralizer of whose Sylow 2-subgroup contains no elements of odd order greater than 3 [1], and also in the casesDenote by δ the variable taking the values + and −. Denote by L δ 3 (p n ) the group L 3 (p n ) if δ = + and the group U 3 (p n ) if δ = −.

Groups Whose Element Orders do not Exceed 6

“…Then N G (V ) = H. If in this case S is not a Sylow 2-subgroup of G, then N G (S) is a dihedral or semidihedral group of order 16, and hence it contains an element of order 8. Therefore, S is a Sylow 2-subgroup of G. In view of [17,Prop. 6], all Sylow 2-subgroups of G are finite and conjugate.…”

Recognizing M10 by spectrum in the class of all groups

“…By Shunkov's theorem [21], H is an infinite 2-group of period 8. By [14,Proposition 5], any finite subgroup of an infinite 2-group is different from its normalizer, in particular, for any finite subgroup K of H, there is a subgroup of an arbitrary large order in H, which contains K.…”