Groups containing a strongly embedded subgroup

An involution v of a group G is said to be finite (in G) if vv ^g has finite order for any g ∈ G. A subgroup B of G is called a strongly embedded (in G) subgroup if B and G\B contain involutions, but B ∩ B ^g does not, for any g ∈ G\B. We prove the following results. Let a group G contain a finite involution and an involution whose centralizer in G is periodic. If every finite subgroup of G of even order is contained in a simple subgroup isomorphic, for some m, to L ₂(2 ^m ) or Sz(2 ^m ), then G is isomorphic to L ₂(Q) or Sz(Q) for some locally finite field Q of characteristic two. In particular, G is locally finite (Thm. 1). Let a group G contain a finite involution and a strongly embedded subgroup. If the centralizer of some involution in G is a 2-group, and every finite subgroup of even order in G is contained in a finite non-Abelian simple subgroup of G, then G is isomorphic to L ₂(Q) or Sz(Q) for some locally finite field Q of characteristic two (Thm. 2). Supported by RFBR (project No. 08-01-00322), by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-334.2008.1), and by the Russian Ministry of Education through the Analytical Departmental Target Program (ADTP) “Development of Scientific Potential of the Higher School of Learning” (project Nos. 2.1.1.419 and 2.1.1./3023). (D. V. Lytkina and V. D. Mazurov) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 190–202, March–April, 2009.