Decomposition of a Group over an Abelian Normal Subgroup
Let a group G have an Abelian normal subgroup A ; put G ¯ = G/A and g ¯ = gA for g ∈ G . We can think of A as a right ℤ G ¯ -module and define the action of an element u = α 1 g ¯ 1 +…+ α n g ¯ n ∈ ℤ G ¯ on a ∈ A by a formula a u = a g 1 α 1 ·…· a g n α n ; here a g i = g i − 1 a g i . Denote by Θ ℤ...
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| Veröffentlicht in: | Algebra and logic 2016-09, Vol.55 (4), p.315-326 |
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| container_title | Algebra and logic |
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| creator | Romanovskii, N. S. |
| description | Let a group
G
have an Abelian normal subgroup
A
; put
G
¯
= G/A and
g
¯
=
gA
for
g
∈
G
. We can think of
A
as a right ℤ
G
¯
-module and define the action of an element
u
=
α
1
g
¯
1
+…+
α
n
g
¯
n
∈ ℤ
G
¯
on
a
∈
A
by a formula
a
u
=
a
g
1
α
1
·…·
a
g
n
α
n
; here
a
g
i
=
g
i
−
1
a
g
i
. Denote by
Θ
ℤ
G
¯
(
A
) the annihilator of
A
in the ring ℤ
G
¯
, which is a two-sided ideal. Let
R
=
ℤ
G
¯
/
Θ
ℤ
G
¯
A
. A subgroup
A
can also be treated as an
R
-module. We give a criterion for the existence of an
R
-decomposition of
G
over
A
, i.e., the possibility of embedding
G
in a semidirect product
G
¯
·
D
, where
D
is an
R
-module. It is also proved that an
R
-decomposition always exists in one important case. |
| doi_str_mv | 10.1007/s10469-016-9401-x |
| format | Article |
| fullrecord | <record><control><sourceid>gale_proqu</sourceid><recordid>TN_cdi_proquest_journals_1880837149</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><galeid>A498802338</galeid><sourcerecordid>A498802338</sourcerecordid><originalsourceid>FETCH-LOGICAL-c453t-34bc180f0ad53f6a2c3f5556490af4fd9fc1a2c1e55207504d4b99a1d6f2de593</originalsourceid><addsrcrecordid>eNp1kM9LwzAUx4MoOKd_gLeC586XX21zHFOnMPSgnkOaJqOjbWrSyvzvzaiHCUoOj7x8Pu-FL0LXGBYYIL8NGFgmUsBZKhjgdH-CZpjnNC0okFM0AwCSckLJOboIYRevIitghoo7o13bu1APtesSZxOVrL0b-8R9Gp-oLlmWpqljfXa-VU3yOpbbw_slOrOqCebqp87R-8P92-ox3bysn1bLTaoZp0NKWalxARZUxanNFNHUcs4zJkBZZithNY5NbDgnkHNgFSuFULjKLKkMF3SObqa5vXcfowmD3LnRd3GlxEUBBc0xO6K2qjGy7qwbvNJtHbRcMhE5QmkRqcUfVDyVaWvtOmPr2P8l4EnQ3oXgjZW9r1vlvyQGechdTrnLmLs85C730SGTEyLbbY0_-vC_0jf5sIMK</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1880837149</pqid></control><display><type>article</type><title>Decomposition of a Group over an Abelian Normal Subgroup</title><source>SpringerLink Journals</source><creator>Romanovskii, N. S.</creator><creatorcontrib>Romanovskii, N. S.</creatorcontrib><description>Let a group
G
have an Abelian normal subgroup
A
; put
G
¯
= G/A and
g
¯
=
gA
for
g
∈
G
. We can think of
A
as a right ℤ
G
¯
-module and define the action of an element
u
=
α
1
g
¯
1
+…+
α
n
g
¯
n
∈ ℤ
G
¯
on
a
∈
A
by a formula
a
u
=
a
g
1
α
1
·…·
a
g
n
α
n
; here
a
g
i
=
g
i
−
1
a
g
i
. Denote by
Θ
ℤ
G
¯
(
A
) the annihilator of
A
in the ring ℤ
G
¯
, which is a two-sided ideal. Let
R
=
ℤ
G
¯
/
Θ
ℤ
G
¯
A
. A subgroup
A
can also be treated as an
R
-module. We give a criterion for the existence of an
R
-decomposition of
G
over
A
, i.e., the possibility of embedding
G
in a semidirect product
G
¯
·
D
, where
D
is an
R
-module. It is also proved that an
R
-decomposition always exists in one important case.</description><identifier>ISSN: 0002-5232</identifier><identifier>EISSN: 1573-8302</identifier><identifier>DOI: 10.1007/s10469-016-9401-x</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Abelian groups ; Algebra ; Analysis ; Decomposition ; Mathematical Logic and Foundations ; Mathematics ; Mathematics and Statistics ; Topological groups ; Uranium</subject><ispartof>Algebra and logic, 2016-09, Vol.55 (4), p.315-326</ispartof><rights>Springer Science+Business Media New York 2016</rights><rights>COPYRIGHT 2016 Springer</rights><rights>Copyright Springer Science & Business Media 2016</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c453t-34bc180f0ad53f6a2c3f5556490af4fd9fc1a2c1e55207504d4b99a1d6f2de593</citedby><cites>FETCH-LOGICAL-c453t-34bc180f0ad53f6a2c3f5556490af4fd9fc1a2c1e55207504d4b99a1d6f2de593</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10469-016-9401-x$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10469-016-9401-x$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,777,781,27905,27906,41469,42538,51300</link.rule.ids></links><search><creatorcontrib>Romanovskii, N. S.</creatorcontrib><title>Decomposition of a Group over an Abelian Normal Subgroup</title><title>Algebra and logic</title><addtitle>Algebra Logic</addtitle><description>Let a group
G
have an Abelian normal subgroup
A
; put
G
¯
= G/A and
g
¯
=
gA
for
g
∈
G
. We can think of
A
as a right ℤ
G
¯
-module and define the action of an element
u
=
α
1
g
¯
1
+…+
α
n
g
¯
n
∈ ℤ
G
¯
on
a
∈
A
by a formula
a
u
=
a
g
1
α
1
·…·
a
g
n
α
n
; here
a
g
i
=
g
i
−
1
a
g
i
. Denote by
Θ
ℤ
G
¯
(
A
) the annihilator of
A
in the ring ℤ
G
¯
, which is a two-sided ideal. Let
R
=
ℤ
G
¯
/
Θ
ℤ
G
¯
A
. A subgroup
A
can also be treated as an
R
-module. We give a criterion for the existence of an
R
-decomposition of
G
over
A
, i.e., the possibility of embedding
G
in a semidirect product
G
¯
·
D
, where
D
is an
R
-module. It is also proved that an
R
-decomposition always exists in one important case.</description><subject>Abelian groups</subject><subject>Algebra</subject><subject>Analysis</subject><subject>Decomposition</subject><subject>Mathematical Logic and Foundations</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Topological groups</subject><subject>Uranium</subject><issn>0002-5232</issn><issn>1573-8302</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp1kM9LwzAUx4MoOKd_gLeC586XX21zHFOnMPSgnkOaJqOjbWrSyvzvzaiHCUoOj7x8Pu-FL0LXGBYYIL8NGFgmUsBZKhjgdH-CZpjnNC0okFM0AwCSckLJOboIYRevIitghoo7o13bu1APtesSZxOVrL0b-8R9Gp-oLlmWpqljfXa-VU3yOpbbw_slOrOqCebqp87R-8P92-ox3bysn1bLTaoZp0NKWalxARZUxanNFNHUcs4zJkBZZithNY5NbDgnkHNgFSuFULjKLKkMF3SObqa5vXcfowmD3LnRd3GlxEUBBc0xO6K2qjGy7qwbvNJtHbRcMhE5QmkRqcUfVDyVaWvtOmPr2P8l4EnQ3oXgjZW9r1vlvyQGechdTrnLmLs85C730SGTEyLbbY0_-vC_0jf5sIMK</recordid><startdate>20160901</startdate><enddate>20160901</enddate><creator>Romanovskii, N. S.</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20160901</creationdate><title>Decomposition of a Group over an Abelian Normal Subgroup</title><author>Romanovskii, N. S.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c453t-34bc180f0ad53f6a2c3f5556490af4fd9fc1a2c1e55207504d4b99a1d6f2de593</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Abelian groups</topic><topic>Algebra</topic><topic>Analysis</topic><topic>Decomposition</topic><topic>Mathematical Logic and Foundations</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Topological groups</topic><topic>Uranium</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Romanovskii, N. S.</creatorcontrib><collection>CrossRef</collection><jtitle>Algebra and logic</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Romanovskii, N. S.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Decomposition of a Group over an Abelian Normal Subgroup</atitle><jtitle>Algebra and logic</jtitle><stitle>Algebra Logic</stitle><date>2016-09-01</date><risdate>2016</risdate><volume>55</volume><issue>4</issue><spage>315</spage><epage>326</epage><pages>315-326</pages><issn>0002-5232</issn><eissn>1573-8302</eissn><abstract>Let a group
G
have an Abelian normal subgroup
A
; put
G
¯
= G/A and
g
¯
=
gA
for
g
∈
G
. We can think of
A
as a right ℤ
G
¯
-module and define the action of an element
u
=
α
1
g
¯
1
+…+
α
n
g
¯
n
∈ ℤ
G
¯
on
a
∈
A
by a formula
a
u
=
a
g
1
α
1
·…·
a
g
n
α
n
; here
a
g
i
=
g
i
−
1
a
g
i
. Denote by
Θ
ℤ
G
¯
(
A
) the annihilator of
A
in the ring ℤ
G
¯
, which is a two-sided ideal. Let
R
=
ℤ
G
¯
/
Θ
ℤ
G
¯
A
. A subgroup
A
can also be treated as an
R
-module. We give a criterion for the existence of an
R
-decomposition of
G
over
A
, i.e., the possibility of embedding
G
in a semidirect product
G
¯
·
D
, where
D
is an
R
-module. It is also proved that an
R
-decomposition always exists in one important case.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10469-016-9401-x</doi><tpages>12</tpages></addata></record> |
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| identifier | ISSN: 0002-5232 |
| ispartof | Algebra and logic, 2016-09, Vol.55 (4), p.315-326 |
| issn | 0002-5232 1573-8302 |
| language | eng |
| recordid | cdi_proquest_journals_1880837149 |
| source | SpringerLink Journals |
| subjects | Abelian groups Algebra Analysis Decomposition Mathematical Logic and Foundations Mathematics Mathematics and Statistics Topological groups Uranium |
| title | Decomposition of a Group over an Abelian Normal Subgroup |
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