Some apsects of products of totally permutable groups.

Abstract

In this work groups factorised as products of pairwise totally permutable subgroups are investigated in the framework of SC-groups, P ST -groups, PT -groups and T -groups. In particular an attempt is made to answer the following questions:

Let G = G1G2 ··· Gr be a product of pairwise totally permutable subgroups. If G is a T -group, under what conditions is Gi a T -group, for each i 2{1, 2,r}?

···

Let G = G1G2 ··· Gr be a product of pairwise totally permutable T -groups. Under what conditions is G a T -group?

Similar questions are raised for PT -groups, P ST -groups and SC-groups. It turns out that pairwise totally permutable factors are T -groups whenever the product is a T -group. Similar results hold for PT -groups, P ST -groups and SC-groups. The

product of pairwise totally permutable SC-groups is also an SC-group. However the direct product of P ST -groups need not be a P ST -group. If G = G1G2 ··· Gr is the direct product of soluble P ST -groups with |Gi| and |Gj| co-prime for all

i, j 2{1, 2, ··· ,r} with i =6j, then G is a P ST -group.