## Element orders in coverings of symmetric and alternating groups.(English. Russian original)Zbl 0930.20003

Algebra Logika 38, No. 3, 296-315 (1999); translation in Algebra Logic 38, No. 3, 159-170 (1999).
Which finite groups can be characterized by their element orders? How the element orders are changed under extensions of groups? These questions are very popular in the modern theory of finite groups (see, for example, the survey by W. J. Shi [in Group theory in China, Kluwer Academic Publishers, Math. Appl., Dordr. 365, 163-181 (1996; Zbl 0872.20026)]).
Theorem. Let $$G$$ be a finite group and let $$H$$ be a quotient group of $$G$$ by a nontrivial subgroup. If $$H$$ is isomorphic to a symmetric group or an alternating group of degree $$m\geq 5$$, then $$G$$ contains an element whose order differs from the order of any element of $$H$$.

### MSC:

 20B35 Subgroups of symmetric groups 20D06 Simple groups: alternating groups and groups of Lie type 20D60 Arithmetic and combinatorial problems involving abstract finite groups

Zbl 0872.20026