The Solution of a Problem of Kegel and Wehrfritz

On the Centralizer of an Element of Order Four in a Locally Finite Group

“…Combining this with (5), we deduce that for an arbitrary finite subset T of G there exists a ∈ A such that T ⊆ K a . It follows that γ 3 (G) is nilpotent of class at most c. This contradicts (1). The proof is complete.…”

Fixed Points of Automorphisms of Certain Locally Finite Groups and Chevalley Groups

“…Safiro and Sunkov [20,21], and the author [11]). In particular, the case of our theorem when the automorphism has order 2 was handled by Asar [1], using much less than the full classification of finite simple groups and giving a stronger conclusion, and a result similar to Asar's has been announced independently by Pavljuk [19]. For automorphisms of prime order, our theorem has been proved by Turau [29] using somewhat different methods.…”

“…Since / generates the Galois group of K over K o , any element of A has the f o r m / / ' , where f x is a ^-automorphism of # a n d / h a s been extended to K. Clearly f x {a~l Ea) = a~x Ea, since a is a AT-rational matrix. Also fiaT 1 Ea) = f{a)~1f{E)f{a) = a' 1 M" 1 Eua (using (3.1m)) is equal to a" 1 Ea, since clearly u normalizes E. Similarly, an element a~xxa of a' 1 Ea is ^-rational if and only if it is fixed by all elements of A. If xe<p(G(K)), then a~lxa is ^-rational and so fixed by f x (in the notation just used), while f{a~lxa) = a' 1 u~lj{x) ua = a~lxa, from (3.1g).…”

Involutions in Locally Finite Groups

“…This was proved by Hartley [15] for periodic almost locally soluble groups G while Asar showed that any locally finite group with a Chernikov centralizer of some involution is almost locally soluble [1]. We recall that a group G is Chernikov if it has a subgroup of finite index that is a direct product of finitely many groups of type C p ∞ for various primes p (quasicyclic p-groups).…”