On Finite Groups Subspectral to Finite Almost Simple Groups

Spectrum !(G) of a finite group G is the set of element orders of G. This set is closed under divisibility of its elements, therefore it can be uniquely defined by its subset μ(G) consisting of maximal under divisibility elements of !(G). Two groups are said to be isospectral, if their spectra coincide. A finite group G is called recognizable by spectrum in the class of finite groups (or recognaizable), if every finite group whose spectrum coincides with !(G) is isomorphic to G. In a recent survey dedicated to recognizability of finite groups, an unsolved question is mentioned about recognizability of symmetric group S10 of all permutations of degree 10. Difficulty of this problem is, in particular, due to a wast number of finite simple groups, which are subspectral to S10, i. e. simple groups whose spectra are subsets of !(S10). This paper gives a method of finding all groups subspectral to a given group, and for every alternating group L the list of subspectral to S10 covers of L are given, whose basements are irreducible modules of representations of L over finite fields.