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# SOME SUPERSOLVABILITY CONDITIONS FOR FINITE GROUPS

F. Barry, D. MachHale and À. Ní Shé
Mathematical Proceedings of the Royal Irish Academy
Vol. 106A, No. 2 (December 2006), pp. 163-177
Stable URL: http://www.jstor.org/stable/40656943
Page Count: 15
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## Abstract

To investigate the intuitive notion that the closer a group is to being abelian the more likely it is to be supersolvable, we investigate situations where the following indicators of commutativity are ' large'— the reciprocal of the index |G'|; reciprocal of |G'|; $Pr(G)\, = \,{{K(G)} \over {|G|}}$; $f(G)\, = \,\sum\nolimits_{i = 1}^k {{{di} \over {|G|}}}$ and l(G), the maximum proportion of elements inverted by an automorphism. Applications to groups that satisfy the converse of Lagrange's Theorem are deduced.

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