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On critical groups

Published online by Cambridge University Press:  09 April 2009

L. G. Kovács  and
M. F. Newman
L. G. Kovács
Affiliation:
Australian National University Canberra
M. F. Newman
Affiliation:
Australian National University Canberra
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The concept of critical group was introduced by D. C. Cross (as reported by G. Higman in [5]): a finite group is called critical if it is not contained in the variety generated by its proper factors. (The factors of a group G are the groups H/K where K H ≦ G, and H/K is a proper factor of G unless H = G and K =1). Some investigations concerning finite groups and varieties depend on the investigator's ability to decide whether a given group is critical or not. (For instance, one of the crucial points in the important paper [9] of Sheila Oates and M. B. Powell is a necessary condition of criticality: their Lemma 2.4.2.) An obvious necessary condition is that the group should have only one minimal normal subgroup: the group is then called monolithic, and the minimal normal subgroup its monolith. This is, however, far from being a sufficient condition, and it is the purpose of the present paper to give some sufficient conditions for the criticality of monolithic groups. (We consider the trivial group neither monolithic nor critical.) The basis of our results is an analysis of the following situation.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 6 , Issue 2 , May 1966 , pp. 237 - 250
Copyright
Copyright © Australian Mathematical Society 1966

References

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