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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Hughes conjecture and groups with absolutely regular subgroups or $ ECF$-subgroups

Author: Joseph A. Gallian
Journal: Proc. Amer. Math. Soc. 49 (1975), 315-318
MSC: Primary 20D25
MathSciNet review: 0364442
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Abstract: Let $ G$ be a finite $ p$-group and $ {H_p}(G)$ the subgroup generated by the elements of $ G$ of order different from $ p$. Hughes conjectured that if $ G > {H_p}(G) > 1$, then $ \vert G:{H_p}(G)\vert = p$. Although the conjecture is not true for all $ G$, it is shown here that if $ G$ has a subgroup $ L$ such that $ \vert L:{L^p}\vert \leq {p^{p - r}}(r \geq 1)$ and $ {p^r} \leq \vert G:L\vert \leq {p^{r + p}}$ or an ECF-subgroup $ L$ with $ \vert G:{L_2}\vert \leq {p^{p + 2}}$, then $ G$ satisfies the Hughes conjecture.

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Keywords: Finite $ p$-groups, $ {H_p}$-problem, Hughes conjecture, ECF-groups, absolutely regular groups
Article copyright: © Copyright 1975 American Mathematical Society