ON MINIMAL NON-p-CLOSED GROUPS AND RELATED PROPERTIES

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 and homomorphic images are. In this paper we characterize finite minimal non-p-closed groups and investigate the relationship between them and minimal non-p-exponent closed groups. In particular, we show that every minimal non-p-closed group is non-p-exponent closed and that minimal non-p-closed groups and simple minimal non-p-exponent closed groups have cyclic Sylow p-subgroups. Furthermore, given a prime p, we describe non-p-exponent closed groups of smallest order and we show that they coincide with non-p-closed groups of smallest order. Mathematics Subject Classification (2010). Primary 20D05, 20D25. Secondary 20F99.