The Hughes conjecture and groups with absolutely regular subgroups or $ECF$-subgroups
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- by Joseph A. Gallian PDF
- Proc. Amer. Math. Soc. 49 (1975), 315-318 Request permission
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 $|G:{H_p}(G)| = p$. Although the conjecture is not true for all $G$, it is shown here that if $G$ has a subgroup $L$ such that $|L:{L^p}| \leq {p^{p - r}}(r \geq 1)$ and ${p^r} \leq |G:L| \leq {p^{r + p}}$ or an ECF-subgroup $L$ with $|G:{L_2}| \leq {p^{p + 2}}$, then $G$ satisfies the Hughes conjecture.References
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Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 49 (1975), 315-318
- MSC: Primary 20D25
- DOI: https://doi.org/10.1090/S0002-9939-1975-0364442-0
- MathSciNet review: 0364442