Let p be a prime. A group is called p-closed if it has a normal Sylow p-subgroup and it is called p-exponent closed if the elements of order dividing p form a subgroup. A group is minimal non-p-closed if it is not p-closed but its proper subgroups and homomorphic images are. Similarly, a group is called minimal non-p-exponent closed if it is not p-exponent closed but all its proper subgroups an...

Abstract A classical result of Baer states that a finite group G which is the product two normal supersoluble subgroups if and only ʹ nilpotent. In this article, we show = AB (respectively, w -supersoluble) B , in permutes with every maximal subgroup each Sylow then -supersoluble), provided We also investigate products defined above when $ A\cap B=1 obtain more general results.

The totality formed by all the operators of any group (G) which are common to all the invariant subgroups of prime index (p) constitutes a characteristic subgroup, and the corresponding quotient group is the abelian group of order pK and of type (1, 1, 1, ■■■)-\ The number of the invariant subgroups of index p is therefore pK — 1/p — 1. The given totality includes all the operators of G which a...

Theorem 1.1 Suppose that G is a finite group, and k is a field of characteristic p. Let H be a collection of subgroups of G. Denote by K the collection of subgroups K of G with the property that the Sylow p-subgroups of the centralizer CG(K) are not conjugate to a subgroup of any of the groups in H. Let J be the ideal in H∗(G, k) given by the sum of the images of transfer from subgroups in H, a...