Vol. 27, No. 1, 1968

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Maximal nonnormal chains in finite groups

Armond E. Spencer

Vol. 27 (1968), No. 1, 167–173

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
Received: 27 October 1967
Published: 1 October 1968
Armond E. Spencer