Two nonsimplicity criteria for groups with an infinitely isolated subgroup

Abstract

The concept of an infinitely isolated subgroup was first introduced in [3] in connection with the abstract characterization of groups of type P~ (~K) over a locally finite field K of odd characteristic.It played a very important role in the solution of S. N.Chernikov's minimality problem [22,25] and occupies a central position in the structure of the theory of locally finite groups with various finiteness conditions [15,20,26].The solution of S. N. Chernikov's minimality problem in other classes of periodic groups [27][28][29] also required consideration of periodic groups with an infinitely isolated subgroup.As in the case of locally finite groups [22], here it is necessary to consider the situation where an infinitely isolated subgroup is at the same time strongly embedded. In the present paper this situation is studied "in its pure form," i.e., we consider the class of periodic groups with a strongly embedded, infinitely isolated subgroups. It is shown by examples that this class of groups is rather broad and includes groups that are not locally finite. A periodic group with a strongly embedded, infinitely isolated subgroup can be either simple or nonsimple. The authors have found conditions under which such a group has an abelian normal subgroup (Theorems i and 2). These nonsimplicity criteria have applications to the study of groups with various finiteness conditions.The main ideas of this paper and also the proof of Theorem 2 are due to V. P. Shunkov, and the proofs of Theorem i and Lemmas 1-4 to A. N. Izmailov; Lemma 5 was proved jointly. i. Definitions and Known ResultsWe will refer to the definitions and known results stated in this section as propositions with the appropriate number. i. A (finite or infinite group) will be called a group of even order if it contains Involutions and a group of odd order if it contains no involutions. 2. A group is called a Chernikov group if it is a finite extension of a direct product of a finite number of quasicyclic subgroups [i]. 3. A group is called weakly conjugately biprimitively finite if any two conjugate elements of prime order generate a finite subgroup.

Groups with handles of order different from three

“…Hence, C a L G ( ) ∩ = 1, and L l 〈 a 〉 is a Frobenius group. By virtue of the Podufalov lemma [8] and the results of [11], this yields…”

“…Since 〈 a 〉 is a p-finite handle and B l 〈 a 〉 is a locally finite p-group, we conclude that C a B a l 〈 〉 ( ) < ∞ (see, e.g., [8]), i.e., for any element r ∈ B, we have ra = a h for a certain h ∈ B. Taking into account that, as proved above, ( G, B, a h ) ∈ ᑣ and B < H, we complete the proof of assertion (vi).…”

“…Assume that P = P / B is an infinite group ( B P by virtue of the maximality of P and H and the fact that B is a locally finite group normal in H ) . It follows from [8] that C a P ( ) =…”

“…Since the elements a and at are conjugate in B l 〈 a 〉 (see, e.g., [8]), we conclude that the subgroups K g = 〈 〉 at a g , , g ∈ F H \ 1 , are finite [condition (ii) and Lemma 1], K g has the form K g = D g l Z, where Z is an elementary Abelian p-group of order p 2 , and at ∈ Z. According to Lemma 1, cyclic subgroups of the form 〈 ah 〉, h ∈ S, are handles of an Mp-group H , and, therefore, a subgroup of the form F ar l 〈 〉, r ∈ S , is also a Frobenius group with noninvariant factor 〈 〉 ar .…”

“…In L, we choose an element t ≠ 1 and S = 〈 t 〉 × 〈 a 〉. Consider the subgroup B 1 S. It follows from the results of [8,13] that S contains a nonunique element r such that C G ( r ) ∩ B 1 is an infinite group. If r = ha, where h ∈ 〈 t 〉 < B, then, by virtue of Lemma 1 and condition (iii) of Theorem 1, the intersection H ∩ C G ( r ) ∩ B 1 is infinite.…”