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Finite groups with short nonnormal chains

Published online by Cambridge University Press:  09 April 2009

Armond E. Spencer
Armond E. Spencer
Affiliation:
State University of New York, Potsdam, New York, 13676, U.S.A.
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This note is a continuation of the author's work [6], describing the structure of a finite group given some information about the distribution of the subnormal subgroups in the lattice of all subgoups.

DEFINITION. An upper chain of length n in the finite group G is a sequence of subgroups of G; G = Go > G1 > … > Gn, such that for each i, Gi is a maximal subgroup of Gi-1. Let h(G) = n if every upper chain in G of length n contains a proper ( ≠ G) subnormal entry, and there is at least one upper chain in G of length (n – 1) which contains no proper subnormal entry.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 18 , Issue 1 , August 1974 , pp. 111 - 118
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Burnside, W., Theory of groups of finite order, 2nd ed. (Dover, New York, 1955).Google Scholar
[2]Deskins, W. E., ‘A condition for the solvability of a finite group’, Illinois J. Math. 2 (1961), 306–313.Google Scholar
[3]Gorenstein, D., Finite Groups (Harper and Row, New York, 1968).Google Scholar
[4]Janko, Z., ‘Finite groups with invariant fourth maximal subgroups’, Math. Zeit. 82 (1963), 82–89.Google Scholar
[5]Mann, H., ‘Finite groups whose n-maximal subgroups are submnormal’, Trans. Amer. Math. Soc. 2 (1968), 395–409.Google Scholar
[6]Spencer, A., ‘Maximal nonnormal chains in finite groups’, Pacific J. Math. 27 (1968), 167–173.Google Scholar