KIVANC¸ ERSOY. This is a joint-work with Pavel Shumyatsky. We study the
following question: Problem. Does there exist an infinite simple locally finite
group G.
FIXED POINTS OF AUTOMORPHISMS IN SIMPLE LOCALLY FINITE GROUPS KIVANC ¸ ERSOY
This is a joint-work with Pavel Shumyatsky. We study the following question: Problem. Does there exist an infinite simple locally finite group G with an elementary abelian p-group of automorphisms A of order pr with r ≥ 3 satisfying the following conditions: (1) CG (A) is finite, (2) For every α ∈ A\{1}, the subgroup (CG (α))0 has finite exponent? In this talk, we will answer this question negatively. In particular, we will prove the following results: Theorem 1. (E.-Shumyatsky) Let G be a simple locally finite group and A be an elementary abelian p-group of automorphisms of G of order pr such that (1) CG (A) is finite, (2) There exists some d ≥ 0 such that 2d ≤ r − 1 and for every α ∈ A\{1}, the subgroup (CG (α))(d) has finite exponent. Then G is finite. Theorem 2. (E.-Shumyatsky) Let G be an infinite simple locally finite group and A be an elementary abelian p−subgroup of automorphisms of G such that (1) CG (A) is finite. (2) For every α ∈ A\{1}, (CG (x))(d) has finite exponent for some d ≥ 2. Then G ∼ = P SLp (k) for some infinite locally finite field k of characteristic q 6= p, the subgroup A consists of inner-diagonal automorphisms and |A| = p2 . Department of Mathematics, Mimar Sinan Fine Arts University Istanbul, Turkey E-mail address:
[email protected]
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