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Finite-by-Nilpotent Groups and a Variation of the BFC-Theorem
For a group G and an element a ∈ G , let | a | k denote the cardinality of the set of commutators [ a , x 1 , ⋯ , x k ] , where x 1 , ⋯ , x k range over G . The main result of the paper states that a group G is finite-by-nilpotent if and only if there are positive integers k and n , such that | x |...
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| Published in: | Mediterranean journal of mathematics 2022-10, Vol.19 (5), Article 202 |
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| Format: | Article |
| Language: | English |
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| cites | cdi_FETCH-LOGICAL-c319t-d8f6a2c57b807da86d0ac6a1b3d33281f6490a61c98e54d5bc240a4d7b5f7f503 |
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| container_issue | 5 |
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| container_title | Mediterranean journal of mathematics |
| container_volume | 19 |
| creator | Shumyatsky, Pavel |
| description | For a group
G
and an element
a
∈
G
, let
|
a
|
k
denote the cardinality of the set of commutators
[
a
,
x
1
,
⋯
,
x
k
]
, where
x
1
,
⋯
,
x
k
range over
G
. The main result of the paper states that a group
G
is finite-by-nilpotent if and only if there are positive integers
k
and
n
, such that
|
x
|
k
≤
n
for every
x
∈
G
. More precisely, if
|
x
|
k
≤
n
for every
x
∈
G
, then
γ
k
+
1
(
G
)
has finite (
k
,
n
)-bounded order. Furthermore, in any group
G
, the set
F
C
k
(
G
)
=
{
x
∈
G
;
|
x
|
k
<
∞
}
is a subgroup and
γ
k
+
1
(
F
C
k
(
G
)
)
is locally normal. |
| doi_str_mv | 10.1007/s00009-022-02140-0 |
| format | article |
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G
and an element
a
∈
G
, let
|
a
|
k
denote the cardinality of the set of commutators
[
a
,
x
1
,
⋯
,
x
k
]
, where
x
1
,
⋯
,
x
k
range over
G
. The main result of the paper states that a group
G
is finite-by-nilpotent if and only if there are positive integers
k
and
n
, such that
|
x
|
k
≤
n
for every
x
∈
G
. More precisely, if
|
x
|
k
≤
n
for every
x
∈
G
, then
γ
k
+
1
(
G
)
has finite (
k
,
n
)-bounded order. Furthermore, in any group
G
, the set
F
C
k
(
G
)
=
{
x
∈
G
;
|
x
|
k
<
∞
}
is a subgroup and
γ
k
+
1
(
F
C
k
(
G
)
)
is locally normal.</description><identifier>ISSN: 1660-5446</identifier><identifier>EISSN: 1660-5454</identifier><identifier>DOI: 10.1007/s00009-022-02140-0</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Commutators ; Mathematics ; Mathematics and Statistics ; Subgroups</subject><ispartof>Mediterranean journal of mathematics, 2022-10, Vol.19 (5), Article 202</ispartof><rights>The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022. Springer Nature or its licensor 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><citedby>FETCH-LOGICAL-c319t-d8f6a2c57b807da86d0ac6a1b3d33281f6490a61c98e54d5bc240a4d7b5f7f503</citedby><cites>FETCH-LOGICAL-c319t-d8f6a2c57b807da86d0ac6a1b3d33281f6490a61c98e54d5bc240a4d7b5f7f503</cites><orcidid>0000-0002-4976-5675</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Shumyatsky, Pavel</creatorcontrib><title>Finite-by-Nilpotent Groups and a Variation of the BFC-Theorem</title><title>Mediterranean journal of mathematics</title><addtitle>Mediterr. J. Math</addtitle><description>For a group
G
and an element
a
∈
G
, let
|
a
|
k
denote the cardinality of the set of commutators
[
a
,
x
1
,
⋯
,
x
k
]
, where
x
1
,
⋯
,
x
k
range over
G
. The main result of the paper states that a group
G
is finite-by-nilpotent if and only if there are positive integers
k
and
n
, such that
|
x
|
k
≤
n
for every
x
∈
G
. More precisely, if
|
x
|
k
≤
n
for every
x
∈
G
, then
γ
k
+
1
(
G
)
has finite (
k
,
n
)-bounded order. Furthermore, in any group
G
, the set
F
C
k
(
G
)
=
{
x
∈
G
;
|
x
|
k
<
∞
}
is a subgroup and
γ
k
+
1
(
F
C
k
(
G
)
)
is locally normal.</description><subject>Commutators</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Subgroups</subject><issn>1660-5446</issn><issn>1660-5454</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kD1PwzAQhi0EEuXjDzBZYjacHceOBwaoaEGqYCmslhPbNFUbB9sd-u9JCYKNk053w_shPQhdUbihAPI2wTCKAGPDUg4EjtCECgGk5CU__v25OEVnKa0BmKIFm6C7Wdu12ZF6T17aTR-y6zKex7DrEzadxQa_m9ia3IYOB4_zyuGH2ZQsVy5Et71AJ95skrv8uefobfa4nD6Rxev8eXq_IE1BVSa28sKwppR1BdKaSlgwjTC0LmxRsIp6wRUYQRtVuZLbsm4YB8OtrEsvfQnFOboec_sYPncuZb0Ou9gNlZpJkJVSHOSgYqOqiSGl6LzuY7s1ca8p6AMmPWLSAyb9jUkfoovRlAZx9-HiX_Q_ri8Oz2iX</recordid><startdate>20221001</startdate><enddate>20221001</enddate><creator>Shumyatsky, Pavel</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-4976-5675</orcidid></search><sort><creationdate>20221001</creationdate><title>Finite-by-Nilpotent Groups and a Variation of the BFC-Theorem</title><author>Shumyatsky, Pavel</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c319t-d8f6a2c57b807da86d0ac6a1b3d33281f6490a61c98e54d5bc240a4d7b5f7f503</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Commutators</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Subgroups</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Shumyatsky, Pavel</creatorcontrib><collection>CrossRef</collection><jtitle>Mediterranean journal of mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Shumyatsky, Pavel</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Finite-by-Nilpotent Groups and a Variation of the BFC-Theorem</atitle><jtitle>Mediterranean journal of mathematics</jtitle><stitle>Mediterr. J. Math</stitle><date>2022-10-01</date><risdate>2022</risdate><volume>19</volume><issue>5</issue><artnum>202</artnum><issn>1660-5446</issn><eissn>1660-5454</eissn><abstract>For a group
G
and an element
a
∈
G
, let
|
a
|
k
denote the cardinality of the set of commutators
[
a
,
x
1
,
⋯
,
x
k
]
, where
x
1
,
⋯
,
x
k
range over
G
. The main result of the paper states that a group
G
is finite-by-nilpotent if and only if there are positive integers
k
and
n
, such that
|
x
|
k
≤
n
for every
x
∈
G
. More precisely, if
|
x
|
k
≤
n
for every
x
∈
G
, then
γ
k
+
1
(
G
)
has finite (
k
,
n
)-bounded order. Furthermore, in any group
G
, the set
F
C
k
(
G
)
=
{
x
∈
G
;
|
x
|
k
<
∞
}
is a subgroup and
γ
k
+
1
(
F
C
k
(
G
)
)
is locally normal.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s00009-022-02140-0</doi><orcidid>https://orcid.org/0000-0002-4976-5675</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1660-5446 |
| ispartof | Mediterranean journal of mathematics, 2022-10, Vol.19 (5), Article 202 |
| issn | 1660-5446 1660-5454 |
| language | eng |
| recordid | cdi_proquest_journals_2707899407 |
| source | Springer Nature |
| subjects | Commutators Mathematics Mathematics and Statistics Subgroups |
| title | Finite-by-Nilpotent Groups and a Variation of the BFC-Theorem |
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