English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In this paper the Hughes conjecture for finite pgroups is proved in the case of groups which are nilpotent of class 2p-2. If G is a group and p is a prime number then the Hughes subgroup HpiG) of G is by definition the subgroup generated by all elements of G that do not have order p. The conjecture of Hughes was that if G>HpiG)>l then HviG) must have index p in G; see [o]. Hughes showed that the conjecture is true for p = 2 and all G, and Straus and Szekeres [lO] proved it for p = 3 and all G. Hughes and Thompson [7] proved the corresponding result for all p and for G finite and nonnilpotent. The most interesting cases which remain are those in which G is a finite 7^-group with p ^ 5 ; throughout this note we shall take G to be a finite £-group, and our results will be of most significance when pit5. We may remark at this point that it is a nontrival exercise to construct a £-group GP for each prime p such that Gp>HPiGp) > 1. Though Wall [ll] has shown by means of an example that the Hughes conjecture is false for finite 5-groups and p — 5,it is nevertheless of some interest to find sufficient conditions for the conjecture to hold. Thus Zappa [12] has shown that all is well when G is nilpotent of class p. The main result of the present note is a substantial strengthening of this: Theorem. The Hughes conjecture is valid for finite p-groups of nilpotency class 2p — 2. It seems likely to us that class 2p — 2 cannot be replaced by a much greater class without falsifying the theorem. The proof of this theorem depends heavily on the main result of [8], which we state again in the interests of clarity: Theorem [8]. If \G:HPiG)\^p2, V„(G)£t(G) and y2p(G) = l, thenVPiHpiG)r\y2iG)) = l. Received by the editors February 3, 1970. AMS 1970 subject classifications. Primary 20D15, 20D25.

Solution of the Hughes problem for finite 𝑝-groups of class 2𝑝-2.