Old, Recent and New Results on Quasinormal Subgroups

Thus normal subgroups are always quasinormal, but not conversely. For, if p is a prime, then any cyclic group Cpn extended by any cyclic group Cpm has all subgroups quasinormal (provided, when p = 2 and n > 2, the cyclic subgroup of order 4 in C2n is central in the extension). The same is true if Cpn is replaced by any abelian p-group H of finite exponent, with Cpm acting on H as a group of power automorphisms (and elements of order 4 in H are again central in the extension if p = 2). These results can be found in sections 2.3 and 2.4 of [16]. One of the earliest results about quasinormal subgroups is due to Ore, who proved in 1938 that a quasinormal subgroup of a finite group G is always subnormal in G ([14]). When G is infinite, then A does not have to be subnormal, but it is always ascendant in G (see [17]). Clearly the extent to which a quasinormal subgroup A can differ from being normal is of interest and a measure of this was given by Ito and Szep in 1962 when they proved that, again with G finite, and denoting the core of A in G by AG, the quotient A/AG is always nilpotent