FINITE GROUPS WITH A CYCLIC NORM QUOTIENT

Abstract

The norm N(G) of a group G is the intersection of the nor- malizers of all the subgroups of G. In this paper, the structure of finite groups with a cyclic norm quotient is determined. As an application of the result, an interesting characteristic of cyclic groups is given, which as- serts that a finite group G is cyclic if and only if Aut(G)/P(G) is cyclic, where P(G) is the power automorphism group of G. The norm N(G) of a group G, first introduced by R. Baer (1) in 1934, is the intersection of the normalizers of all the subgroups of G. It is clear that N(G) is a characteristic subgroup of G and it contains the center Z(G). Also N(G) itself is a Dedekind group, and every element of N(G) induces a power automorphism on G. Many authors have investigated both N(G) and how N(G) influences the structure of G (see (2), (3), (4), (9), (10) and (11)). For instance, Schenkman proved that N(G) ≤ Z2(G) for any group G (see (9)), and Baer showed that a 2-group G must be a Dedekind group if the N(G) is nonabelian (see (2)). In (10) and (11), the authors have determined the structure of a finite group G satisfying |G : N(G)| = p or pq, where p,q are primes. In this paper, finite groups with a cyclic norm quotient are determined, see Section 2. Based on the above result, we will establish a characteristic of cyclic groups. Recall that a power automorphism of a group G is an automorphism that leaves every subgroup of G invariant. Under such an automorphism, every element of G is mapped to one of its powers. All power automorphisms of G constitute an abelian normal subgroup, denoted by P(G), of Aut(G). The structure of G and P(G) are strictly linked (for example, see (5), (12)), especially the quotient group Aut(G)/P(G) strongly influences the structure of G. In (11), groups satisfying |Aut(G)/P(G)| = 1,p or pq are completely clarified. In Section 3,