On the structure of the noninvariant factor in some Frobenius groups

Some groups of exponent 72

“…Note also that there exist groups G with .G/ D ¹2; 3 n º that are not locally finite, where n > 7 (see [3,14]), and .G/ D ¹2 m ; 3º, where m > 54 (see [9]). …”

Fixed-point-free 2-finite automorphism groups

“…According to Zhurtov [16,Lemma 8], G contains at most one subgroup isomorphic to SL (2,5); this follows also from [15,Theorem 1] or [9,Theorem 1]. Hence the subgroup T generated by all copies of SL (2,5) and all involutions in G is finite and normal in G. Below we show that the quotient Γ := G/T is locally finite (which implies that G is locally finite).…”

“…For groups G as in Theorem 1, the subgroups generated by all elements of prime order are characterized in Sozutov [9,Theorem 1]. Our theorem does not readily follow from Theorem 3 in [9], since that Theorem 3 imposes a weak version of 2-finiteness on the whole Frobenius group, and not just on the Frobenius complement. …”

A characterization of alternating groups

“…In [3], the theorem above was extended from finite groups to groups in which every two conjugate elements of prime order generate a finite group. Tangible progress was made in [4,5], by showing that a periodic group, which is generated by elements of order 3 and acts freely on a non-trivial Abelian group, is finite.…”

“…The methods developed in the above-cited papers made it possible to give a short and simple proof of the Zassenhaus theorem without using the character theory (see [8]). Finally, the above-mentioned result of [3] was generalized in [9] to the case where the condition of acting freely for a group is imposed only on groups generated by two conjugate elements of prime order. The next result significantly extends the class of groups under investigation by weakening a condition imposed on 2-generated subgroups.…”