Let D be a division ring with center F and assume that N is a locally soluble almost subnormal subgroup of the multiplicative group D,of D. We prove that if N is algebraic over F, then N is central. This answers partially [11, Conjecture 1].

The set of element orders of a nite group G is called the spectrum. groups with coinciding spectra are said to be isospectral. It is known that if G has a nontrivial normal soluble subgroup then there exist in nitely many pairwise non-isomorphic groups isospectral to G. The situation is quite different if G is a nonabelain simple group. Recently it was proved that ifL is a simple classical group of dimension at least 62 and G is a nite group isospectral to L, then up to isomorphism L£G£ Aut L. We show that the assertion remains true if 62 is replaced by 38.

Please click on PDF to view the abstract.

Let L: =U 3 (11). In this article, we classify groups with the same order and degree pattern as an almost simple group related toL. In fact, we prove that L, L: 2 and L: 3 are OD-characterizable, and L: S3 is 5-fold OD-characterizable.

Let G be a finite group and p (G) be the set of all the prime divisors of |G|. The prime graph of G is a simple graph G (G) whose vertex set is p (G) and two distinct vertices p and q are joined by an edge if and only if G has an element of order pq, and in this case we will write p~q. The degree of p is the number of vertices adjacent to p and is denoted by deg (p). If |G|=p1a1p2a2 … pkak, pi’s different primes, p1<p2<…< pk, then D (G) = (deg (p1), deg (p2), …, deg (pk)) is called the degree pattern of G. A finite group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups S with |G|=|S| and D (G) =D (S). In this paper, we characterize groups with the same order and degree pattern as an almost simple groups related to L3 (25).