p-GROUPS WITH CYCLIC OR GENERALISED QUATERNION HUGHES SUBGROUPS: CLASSIFYING TIDY p-GROUPS

Let G be a p-group for some prime p. Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G with order not equal to p. In this paper, we prove that if the Hughes subgroup of G is cyclic, then G has exponent p or is cyclic or is dihedral. We also prove that if the...

Full description

Saved in:
Bibliographic Details
Published in:Bulletin of the Australian Mathematical Society 2023-12, Vol.108 (3), p.443-448
Main Authors: BEIKE, NICOLAS F., CARLETON, RACHEL, COSTANZO, DAVID G., HEATH, COLIN, LEWIS, MARK L., LU, KAIWEN, PEARCE, JAMIE D.
Format: Article
Language:English
Subjects:
Classification
Hypotheses
Quaternions
Subgroups
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by cdi_FETCH-LOGICAL-c360t-56bb2b7889ae4e9ab69f3f5331e365555bccd4afc5c4664440652a291cae2d923
cites cdi_FETCH-LOGICAL-c360t-56bb2b7889ae4e9ab69f3f5331e365555bccd4afc5c4664440652a291cae2d923
container_end_page 448
container_issue 3
container_start_page 443
container_title Bulletin of the Australian Mathematical Society
container_volume 108
creator BEIKE, NICOLAS F.
CARLETON, RACHEL
COSTANZO, DAVID G.
HEATH, COLIN
LEWIS, MARK L.
LU, KAIWEN
PEARCE, JAMIE D.
description Let G be a p-group for some prime p. Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G with order not equal to p. In this paper, we prove that if the Hughes subgroup of G is cyclic, then G has exponent p or is cyclic or is dihedral. We also prove that if the Hughes subgroup of G is generalised quaternion, then G must be generalised quaternion. With these results in hand, we classify the tidy p-groups.
doi_str_mv 10.1017/S000497272300031X
format article
fullrecord <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2886820117</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><cupid>10_1017_S000497272300031X</cupid><sourcerecordid>2886820117</sourcerecordid><originalsourceid>FETCH-LOGICAL-c360t-56bb2b7889ae4e9ab69f3f5331e365555bccd4afc5c4664440652a291cae2d923</originalsourceid><addsrcrecordid>eNp1kE9Pg0AQxTdGE2v1A3jbxDO6f9gFvCGlsAkBZSFaL2TZLqaNtXVpD357aVrjwTiXmcl7vzfJAHCN0S1G2LuTCCE38IhH6DBR_HICRthjzMGc0lMw2svOXj8HF32_HDbGiD8CrxsnKYv6UcJnUaUwmkWZiGBRwiTO4zLMhIwn8KkOq7jMRZHDtE7SWEJZPxywexhloZRiOhN5AisxmcGfxEtw1qn33lwd-xjU07iKUicrEhGFmaMpR1uH8bYlref7gTKuCVTLg452jFJsKGdDtVrPXdVppl3OXddFnBFFAqyVIfOA0DG4OeRu7PpzZ_pts1zv7MdwsiG-z32CMPYGFz64tF33vTVds7GLlbJfDUbN_oXNnxcODD0yatXaxfzN_Eb_T30D4SBqyw</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2886820117</pqid></control><display><type>article</type><title>p-GROUPS WITH CYCLIC OR GENERALISED QUATERNION HUGHES SUBGROUPS: CLASSIFYING TIDY p-GROUPS</title><source>Cambridge Journals Online</source><creator>BEIKE, NICOLAS F. ; CARLETON, RACHEL ; COSTANZO, DAVID G. ; HEATH, COLIN ; LEWIS, MARK L. ; LU, KAIWEN ; PEARCE, JAMIE D.</creator><creatorcontrib>BEIKE, NICOLAS F. ; CARLETON, RACHEL ; COSTANZO, DAVID G. ; HEATH, COLIN ; LEWIS, MARK L. ; LU, KAIWEN ; PEARCE, JAMIE D.</creatorcontrib><description>Let G be a p-group for some prime p. Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G with order not equal to p. In this paper, we prove that if the Hughes subgroup of G is cyclic, then G has exponent p or is cyclic or is dihedral. We also prove that if the Hughes subgroup of G is generalised quaternion, then G must be generalised quaternion. With these results in hand, we classify the tidy p-groups.</description><identifier>ISSN: 0004-9727</identifier><identifier>EISSN: 1755-1633</identifier><identifier>DOI: 10.1017/S000497272300031X</identifier><language>eng</language><publisher>Cambridge, UK: Cambridge University Press</publisher><subject>Classification ; Hypotheses ; Quaternions ; Subgroups</subject><ispartof>Bulletin of the Australian Mathematical Society, 2023-12, Vol.108 (3), p.443-448</ispartof><rights>The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.</rights><rights>The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc. This work is licensed under the Creative Commons Attribution License This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c360t-56bb2b7889ae4e9ab69f3f5331e365555bccd4afc5c4664440652a291cae2d923</citedby><cites>FETCH-LOGICAL-c360t-56bb2b7889ae4e9ab69f3f5331e365555bccd4afc5c4664440652a291cae2d923</cites><orcidid>0000-0002-8189-3730 ; 0000-0002-1476-7929 ; 0000-0001-9968-9773 ; 0000-0001-9627-6922 ; 0000-0003-4705-8288 ; 0000-0002-0708-9963 ; 0000-0003-4456-7570</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.cambridge.org/core/product/identifier/S000497272300031X/type/journal_article$$EHTML$$P50$$Gcambridge$$Hfree_for_read</linktohtml></links><search><creatorcontrib>BEIKE, NICOLAS F.</creatorcontrib><creatorcontrib>CARLETON, RACHEL</creatorcontrib><creatorcontrib>COSTANZO, DAVID G.</creatorcontrib><creatorcontrib>HEATH, COLIN</creatorcontrib><creatorcontrib>LEWIS, MARK L.</creatorcontrib><creatorcontrib>LU, KAIWEN</creatorcontrib><creatorcontrib>PEARCE, JAMIE D.</creatorcontrib><title>p-GROUPS WITH CYCLIC OR GENERALISED QUATERNION HUGHES SUBGROUPS: CLASSIFYING TIDY p-GROUPS</title><title>Bulletin of the Australian Mathematical Society</title><addtitle>Bull. Aust. Math. Soc</addtitle><description>Let G be a p-group for some prime p. Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G with order not equal to p. In this paper, we prove that if the Hughes subgroup of G is cyclic, then G has exponent p or is cyclic or is dihedral. We also prove that if the Hughes subgroup of G is generalised quaternion, then G must be generalised quaternion. With these results in hand, we classify the tidy p-groups.</description><subject>Classification</subject><subject>Hypotheses</subject><subject>Quaternions</subject><subject>Subgroups</subject><issn>0004-9727</issn><issn>1755-1633</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNp1kE9Pg0AQxTdGE2v1A3jbxDO6f9gFvCGlsAkBZSFaL2TZLqaNtXVpD357aVrjwTiXmcl7vzfJAHCN0S1G2LuTCCE38IhH6DBR_HICRthjzMGc0lMw2svOXj8HF32_HDbGiD8CrxsnKYv6UcJnUaUwmkWZiGBRwiTO4zLMhIwn8KkOq7jMRZHDtE7SWEJZPxywexhloZRiOhN5AisxmcGfxEtw1qn33lwd-xjU07iKUicrEhGFmaMpR1uH8bYlref7gTKuCVTLg452jFJsKGdDtVrPXdVppl3OXddFnBFFAqyVIfOA0DG4OeRu7PpzZ_pts1zv7MdwsiG-z32CMPYGFz64tF33vTVds7GLlbJfDUbN_oXNnxcODD0yatXaxfzN_Eb_T30D4SBqyw</recordid><startdate>20231201</startdate><enddate>20231201</enddate><creator>BEIKE, NICOLAS F.</creator><creator>CARLETON, RACHEL</creator><creator>COSTANZO, DAVID G.</creator><creator>HEATH, COLIN</creator><creator>LEWIS, MARK L.</creator><creator>LU, KAIWEN</creator><creator>PEARCE, JAMIE D.</creator><general>Cambridge University Press</general><scope>IKXGN</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7XB</scope><scope>88I</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>L6V</scope><scope>M2P</scope><scope>M7S</scope><scope>PHGZM</scope><scope>PHGZT</scope><scope>PKEHL</scope><scope>PQEST</scope><scope>PQGLB</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PTHSS</scope><scope>Q9U</scope><orcidid>https://orcid.org/0000-0002-8189-3730</orcidid><orcidid>https://orcid.org/0000-0002-1476-7929</orcidid><orcidid>https://orcid.org/0000-0001-9968-9773</orcidid><orcidid>https://orcid.org/0000-0001-9627-6922</orcidid><orcidid>https://orcid.org/0000-0003-4705-8288</orcidid><orcidid>https://orcid.org/0000-0002-0708-9963</orcidid><orcidid>https://orcid.org/0000-0003-4456-7570</orcidid></search><sort><creationdate>20231201</creationdate><title>p-GROUPS WITH CYCLIC OR GENERALISED QUATERNION HUGHES SUBGROUPS: CLASSIFYING TIDY p-GROUPS</title><author>BEIKE, NICOLAS F. ; CARLETON, RACHEL ; COSTANZO, DAVID G. ; HEATH, COLIN ; LEWIS, MARK L. ; LU, KAIWEN ; PEARCE, JAMIE D.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c360t-56bb2b7889ae4e9ab69f3f5331e365555bccd4afc5c4664440652a291cae2d923</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Classification</topic><topic>Hypotheses</topic><topic>Quaternions</topic><topic>Subgroups</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>BEIKE, NICOLAS F.</creatorcontrib><creatorcontrib>CARLETON, RACHEL</creatorcontrib><creatorcontrib>COSTANZO, DAVID G.</creatorcontrib><creatorcontrib>HEATH, COLIN</creatorcontrib><creatorcontrib>LEWIS, MARK L.</creatorcontrib><creatorcontrib>LU, KAIWEN</creatorcontrib><creatorcontrib>PEARCE, JAMIE D.</creatorcontrib><collection>Cambridge University Press Wholly Gold Open Access Journals</collection><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>Science Database (Alumni Edition)</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>ProQuest Materials Science &amp; Engineering Collection</collection><collection>ProQuest Central (Alumni)</collection><collection>ProQuest Central UK/Ireland</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>ProQuest Central Student</collection><collection>ProQuest SciTech Premium Collection</collection><collection>ProQuest Engineering Collection</collection><collection>Science Database</collection><collection>Engineering Database</collection><collection>ProQuest Central (New)</collection><collection>ProQuest One Academic (New)</collection><collection>ProQuest One Academic Middle East (New)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Applied &amp; Life Sciences</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering collection</collection><collection>ProQuest Central Basic</collection><jtitle>Bulletin of the Australian Mathematical Society</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>BEIKE, NICOLAS F.</au><au>CARLETON, RACHEL</au><au>COSTANZO, DAVID G.</au><au>HEATH, COLIN</au><au>LEWIS, MARK L.</au><au>LU, KAIWEN</au><au>PEARCE, JAMIE D.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>p-GROUPS WITH CYCLIC OR GENERALISED QUATERNION HUGHES SUBGROUPS: CLASSIFYING TIDY p-GROUPS</atitle><jtitle>Bulletin of the Australian Mathematical Society</jtitle><addtitle>Bull. Aust. Math. Soc</addtitle><date>2023-12-01</date><risdate>2023</risdate><volume>108</volume><issue>3</issue><spage>443</spage><epage>448</epage><pages>443-448</pages><issn>0004-9727</issn><eissn>1755-1633</eissn><abstract>Let G be a p-group for some prime p. Recall that the Hughes subgroup of G is the subgroup generated by all of the elements of G with order not equal to p. In this paper, we prove that if the Hughes subgroup of G is cyclic, then G has exponent p or is cyclic or is dihedral. We also prove that if the Hughes subgroup of G is generalised quaternion, then G must be generalised quaternion. With these results in hand, we classify the tidy p-groups.</abstract><cop>Cambridge, UK</cop><pub>Cambridge University Press</pub><doi>10.1017/S000497272300031X</doi><tpages>6</tpages><orcidid>https://orcid.org/0000-0002-8189-3730</orcidid><orcidid>https://orcid.org/0000-0002-1476-7929</orcidid><orcidid>https://orcid.org/0000-0001-9968-9773</orcidid><orcidid>https://orcid.org/0000-0001-9627-6922</orcidid><orcidid>https://orcid.org/0000-0003-4705-8288</orcidid><orcidid>https://orcid.org/0000-0002-0708-9963</orcidid><orcidid>https://orcid.org/0000-0003-4456-7570</orcidid><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier ISSN: 0004-9727
ispartof Bulletin of the Australian Mathematical Society, 2023-12, Vol.108 (3), p.443-448
issn 0004-9727
1755-1633
language eng
recordid cdi_proquest_journals_2886820117
source Cambridge Journals Online
subjects Classification
Hypotheses
Quaternions
Subgroups
title p-GROUPS WITH CYCLIC OR GENERALISED QUATERNION HUGHES SUBGROUPS: CLASSIFYING TIDY p-GROUPS
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-03-08T17%3A51%3A54IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=p-GROUPS%20WITH%20CYCLIC%20OR%20GENERALISED%20QUATERNION%20HUGHES%20SUBGROUPS:%20CLASSIFYING%20TIDY%20p-GROUPS&rft.jtitle=Bulletin%20of%20the%20Australian%20Mathematical%20Society&rft.au=BEIKE,%20NICOLAS%20F.&rft.date=2023-12-01&rft.volume=108&rft.issue=3&rft.spage=443&rft.epage=448&rft.pages=443-448&rft.issn=0004-9727&rft.eissn=1755-1633&rft_id=info:doi/10.1017/S000497272300031X&rft_dat=%3Cproquest_cross%3E2886820117%3C/proquest_cross%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c360t-56bb2b7889ae4e9ab69f3f5331e365555bccd4afc5c4664440652a291cae2d923%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=2886820117&rft_id=info:pmid/&rft_cupid=10_1017_S000497272300031X&rfr_iscdi=true