A criterion for nilpotency in finite groups
For a positive integer n , we denote by π ( n ) the set of prime divisors of n . For a group G and a ∈ G , we denote by o ( a ) the order of the element a . We prove that a finite group G is nilpotent if and only if π ( o ( a b ) ) = π ( o ( a ) o ( b ) ) for all a , b ∈ G of coprime orders.
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| Veröffentlicht in: | Archiv der Mathematik 2024, Vol.122 (1), p.13-16 |
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| container_title | Archiv der Mathematik |
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| creator | Li, Binbin Lu, Jiakuan Pang, Linna Zhang, Boru |
| description | For a positive integer
n
, we denote by
π
(
n
)
the set of prime divisors of
n
. For a group
G
and
a
∈
G
, we denote by
o
(
a
) the order of the element
a
. We prove that a finite group
G
is nilpotent if and only if
π
(
o
(
a
b
)
)
=
π
(
o
(
a
)
o
(
b
)
)
for all
a
,
b
∈
G
of coprime orders. |
| doi_str_mv | 10.1007/s00013-023-01925-3 |
| format | Article |
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n
, we denote by
π
(
n
)
the set of prime divisors of
n
. For a group
G
and
a
∈
G
, we denote by
o
(
a
) the order of the element
a
. We prove that a finite group
G
is nilpotent if and only if
π
(
o
(
a
b
)
)
=
π
(
o
(
a
)
o
(
b
)
)
for all
a
,
b
∈
G
of coprime orders.</description><identifier>ISSN: 0003-889X</identifier><identifier>EISSN: 1420-8938</identifier><identifier>DOI: 10.1007/s00013-023-01925-3</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Group theory ; Mathematics ; Mathematics and Statistics</subject><ispartof>Archiv der Mathematik, 2024, Vol.122 (1), p.13-16</ispartof><rights>Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-d428facdae268ca0b408b1d48f073e6193d53e5be7be6946284a306f43683b883</cites><orcidid>0000-0002-4514-4396</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00013-023-01925-3$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00013-023-01925-3$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Li, Binbin</creatorcontrib><creatorcontrib>Lu, Jiakuan</creatorcontrib><creatorcontrib>Pang, Linna</creatorcontrib><creatorcontrib>Zhang, Boru</creatorcontrib><title>A criterion for nilpotency in finite groups</title><title>Archiv der Mathematik</title><addtitle>Arch. Math</addtitle><description>For a positive integer
n
, we denote by
π
(
n
)
the set of prime divisors of
n
. For a group
G
and
a
∈
G
, we denote by
o
(
a
) the order of the element
a
. We prove that a finite group
G
is nilpotent if and only if
π
(
o
(
a
b
)
)
=
π
(
o
(
a
)
o
(
b
)
)
for all
a
,
b
∈
G
of coprime orders.</description><subject>Group theory</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0003-889X</issn><issn>1420-8938</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LxDAQhoMouK7-AU8FjxKdZNJ0clwWXYUFLwreQj_Spcva1KQ97L83WsGbh2Fg3o-Bh7FrAXcCoLiPACCQg0wjjMw5nrCFUBI4GaRTtkg6ciLzfs4uYtwnt6TCLNjtKqtDN7rQ-T5rfcj67jD40fX1MevSpeuTmO2Cn4Z4yc7a8hDd1e9esrfHh9f1E9--bJ7Xqy2vZQEjb5Sktqyb0klNdQmVAqpEo6iFAp0WBpscXV65onLaKC1JlQi6VagJKyJcspu5dwj-c3JxtHs_hT69tNIIqQujIU8uObvq4GMMrrVD6D7KcLQC7DcUO0OxCYr9gWIxhXAOxWTudy78Vf-T-gKUaGL3</recordid><startdate>2024</startdate><enddate>2024</enddate><creator>Li, Binbin</creator><creator>Lu, Jiakuan</creator><creator>Pang, Linna</creator><creator>Zhang, Boru</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-4514-4396</orcidid></search><sort><creationdate>2024</creationdate><title>A criterion for nilpotency in finite groups</title><author>Li, Binbin ; Lu, Jiakuan ; Pang, Linna ; Zhang, Boru</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-d428facdae268ca0b408b1d48f073e6193d53e5be7be6946284a306f43683b883</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Group theory</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Li, Binbin</creatorcontrib><creatorcontrib>Lu, Jiakuan</creatorcontrib><creatorcontrib>Pang, Linna</creatorcontrib><creatorcontrib>Zhang, Boru</creatorcontrib><collection>CrossRef</collection><jtitle>Archiv der Mathematik</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Li, Binbin</au><au>Lu, Jiakuan</au><au>Pang, Linna</au><au>Zhang, Boru</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A criterion for nilpotency in finite groups</atitle><jtitle>Archiv der Mathematik</jtitle><stitle>Arch. Math</stitle><date>2024</date><risdate>2024</risdate><volume>122</volume><issue>1</issue><spage>13</spage><epage>16</epage><pages>13-16</pages><issn>0003-889X</issn><eissn>1420-8938</eissn><abstract>For a positive integer
n
, we denote by
π
(
n
)
the set of prime divisors of
n
. For a group
G
and
a
∈
G
, we denote by
o
(
a
) the order of the element
a
. We prove that a finite group
G
is nilpotent if and only if
π
(
o
(
a
b
)
)
=
π
(
o
(
a
)
o
(
b
)
)
for all
a
,
b
∈
G
of coprime orders.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s00013-023-01925-3</doi><tpages>4</tpages><orcidid>https://orcid.org/0000-0002-4514-4396</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0003-889X |
| ispartof | Archiv der Mathematik, 2024, Vol.122 (1), p.13-16 |
| issn | 0003-889X 1420-8938 |
| language | eng |
| recordid | cdi_proquest_journals_2912679605 |
| source | Springer Online Journals Complete |
| subjects | Group theory Mathematics Mathematics and Statistics |
| title | A criterion for nilpotency in finite groups |
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