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The supersolvable residual of an -group

Published online by Cambridge University Press:  24 October 2008

Ben Brewster  and
Malcolm Ottaway
Ben Brewster
Affiliation:
State University of New York at Binghamton, New York 13901
Malcolm Ottaway
Affiliation:
State University of New York at Binghamton, New York 13901
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Extract

Let be the class of groups possessing a subgroup of index n for each divisor n of the group order. McLain (7) initiated the formal investigation of and observed that every solvable group is a direct factor of an -group. However, subclasses of provide some interesting problems. Various subclasses of which satisfy other properties were studied by McLain; the upshot being that these classes approach supersolvability. This program was pursued by Humphreys (3) and Humphreys and Johnson (4), among others. In particular, , the largest quotient closed subclass of , was considered in (3) and (4). Humphreys (3) has shown an odd order -group is supersolvable, but provides some non-supersolvable groups. Our motivation is that if S ∈ QR0(S3), the formation generated by S3, and V is a faithful irreducible GF(2) [S]-module, the semidirect product VS ∈ .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 80 , Issue 3 , November 1976 , pp. 447 - 450
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

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