Maximal nonnormal chains in finite groups

Armond E. Spencer

Vol. 27 (1968), No. 1, 167–173

DOI: 10.2140/pjm.1968.27.167

Abstract

In a finite group G, knowledge of the distribution of the subnormal subgroups of G can be used, to some extent, to describe the structure of G. Here we show that if G is a finite nonnilpotent, solvable group such that every upper chain of length n in G contains a proper subnormal entry then:

(1) the nilpotent length of G is less than or equal to n.

(2) |G| has at most n distinct prime divisors, furthermore if |G| has n distinct prime divisors, then G has abelian Sylow subgroups.

(3) if |G| has at least (n − 1) distinct prime divisors, then G is a Sylow Tower Group, for some ordering of the primes.

(4) r(G) ≤ n, where r(G) denotes the minimal number of generators for G.

Mathematical Subject Classification

Primary: 20.25

Milestones

Received: 27 October 1967

Published: 1 October 1968