Commutation semigroups of finite metacyclic groups with trivial centre
We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented G ( m , n , k ) = a , b ; a m = 1 , b n = 1 , a b = a k ( m , n , k ∈ Z + ) with ( m , k - 1 ) = 1 and n = ind m ( k ) , the smallest positive integer for which k n = 1 ( mod m ) ,...
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| Veröffentlicht in: | Semigroup forum 2020-06, Vol.100 (3), p.765-789 |
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| description | We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented
G
(
m
,
n
,
k
)
=
a
,
b
;
a
m
=
1
,
b
n
=
1
,
a
b
=
a
k
(
m
,
n
,
k
∈
Z
+
)
with
(
m
,
k
-
1
)
=
1
and
n
=
ind
m
(
k
)
,
the smallest positive integer for which
k
n
=
1
(
mod
m
)
,
with the conjugate of
a
by
b
written
a
b
=
b
-
1
a
b
.
The
right
and
left commutation semigroups of
G
, denoted
P
(
G
)
and
Λ
(
G
)
,
are the semigroups of mappings generated by
ρ
(
g
)
:
G
→
G
and
λ
(
g
)
:
G
→
G
defined by
(
x
)
ρ
(
g
)
=
[
x
,
g
]
and
(
x
)
λ
(
g
)
=
[
g
,
x
]
,
where the commutator of
g
and
h
is defined as
[
g
,
h
]
=
g
-
1
h
-
1
g
h
.
This paper builds on a previous study of commutation semigroups of dihedral groups conducted by the authors with C. Levy. Here we show that a similar approach can be applied to
G
, a metacyclic group with trivial centre. We give a construction of
P
(
G
)
and
Λ
(
G
)
as unions of
containers
, an idea presented in the previous paper on dihedral groups. In the case that
a
is cyclic of order
p
or
p
2
or its index is prime, we show that both
P
(
G
)
and
Λ
(
G
)
are disjoint unions of maximal containers. In these cases, we give an explicit representation of the elements of each commutation semigroup as well as formulas for their exact orders. Finally, we extend a result of J. Countryman to show that, for
G
(
m
,
n
,
k
) with
m
prime, the condition
P
(
G
)
=
Λ
(
G
)
is equivalent to
P
(
G
)
=
Λ
(
G
)
. |
| doi_str_mv | 10.1007/s00233-020-10097-3 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2405323925</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2405323925</sourcerecordid><originalsourceid>FETCH-LOGICAL-c270t-ee54ae1f09f48027733142b7ba1594ef8d7b80c930e917cb1ef819b36d49e1c13</originalsourceid><addsrcrecordid>eNp9kEFLAzEQhYMoWKt_wNOC5-hMZrdpjlKsCgUveg67abamdDc1ySr990a34M3T8GbeewMfY9cItwgg7yKAIOIggGetJKcTNsGSBBdI8pRNAEhyVCjO2UWMW8gaZjRhy4XvuiHVyfm-iLZzm-CHfSx8W7Sud8kWnU21OZidM8Xx9uXSe5GC-3T1rjC2T8FesrO23kV7dZxT9rZ8eF088dXL4_PifsWNkJC4tVVZW2xBteUchJREWIpGNjVWqrTtfC2bORhFYBVK02BeoWpoti6VRYM0ZTdj7z74j8HGpLd-CH1-qUUJFQlSosouMbpM8DEG2-p9cF0dDhpB__DSIy-deelfXppyiMZQzOZ-Y8Nf9T-pb3T2bZU</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2405323925</pqid></control><display><type>article</type><title>Commutation semigroups of finite metacyclic groups with trivial centre</title><source>SpringerLink Journals - AutoHoldings</source><creator>DeWolf, Darien ; Edmunds, Charles C.</creator><creatorcontrib>DeWolf, Darien ; Edmunds, Charles C.</creatorcontrib><description>We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented
G
(
m
,
n
,
k
)
=
a
,
b
;
a
m
=
1
,
b
n
=
1
,
a
b
=
a
k
(
m
,
n
,
k
∈
Z
+
)
with
(
m
,
k
-
1
)
=
1
and
n
=
ind
m
(
k
)
,
the smallest positive integer for which
k
n
=
1
(
mod
m
)
,
with the conjugate of
a
by
b
written
a
b
=
b
-
1
a
b
.
The
right
and
left commutation semigroups of
G
, denoted
P
(
G
)
and
Λ
(
G
)
,
are the semigroups of mappings generated by
ρ
(
g
)
:
G
→
G
and
λ
(
g
)
:
G
→
G
defined by
(
x
)
ρ
(
g
)
=
[
x
,
g
]
and
(
x
)
λ
(
g
)
=
[
g
,
x
]
,
where the commutator of
g
and
h
is defined as
[
g
,
h
]
=
g
-
1
h
-
1
g
h
.
This paper builds on a previous study of commutation semigroups of dihedral groups conducted by the authors with C. Levy. Here we show that a similar approach can be applied to
G
, a metacyclic group with trivial centre. We give a construction of
P
(
G
)
and
Λ
(
G
)
as unions of
containers
, an idea presented in the previous paper on dihedral groups. In the case that
a
is cyclic of order
p
or
p
2
or its index is prime, we show that both
P
(
G
)
and
Λ
(
G
)
are disjoint unions of maximal containers. In these cases, we give an explicit representation of the elements of each commutation semigroup as well as formulas for their exact orders. Finally, we extend a result of J. Countryman to show that, for
G
(
m
,
n
,
k
) with
m
prime, the condition
P
(
G
)
=
Λ
(
G
)
is equivalent to
P
(
G
)
=
Λ
(
G
)
.</description><identifier>ISSN: 0037-1912</identifier><identifier>EISSN: 1432-2137</identifier><identifier>DOI: 10.1007/s00233-020-10097-3</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algebra ; Commutation ; Commutators ; Containers ; Mathematics ; Mathematics and Statistics ; Research Article ; Unions</subject><ispartof>Semigroup forum, 2020-06, Vol.100 (3), p.765-789</ispartof><rights>Springer Science+Business Media, LLC, part of Springer Nature 2020</rights><rights>Springer Science+Business Media, LLC, part of Springer Nature 2020.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-ee54ae1f09f48027733142b7ba1594ef8d7b80c930e917cb1ef819b36d49e1c13</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/s00233-020-10097-3$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00233-020-10097-3$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27923,27924,41487,42556,51318</link.rule.ids></links><search><creatorcontrib>DeWolf, Darien</creatorcontrib><creatorcontrib>Edmunds, Charles C.</creatorcontrib><title>Commutation semigroups of finite metacyclic groups with trivial centre</title><title>Semigroup forum</title><addtitle>Semigroup Forum</addtitle><description>We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented
G
(
m
,
n
,
k
)
=
a
,
b
;
a
m
=
1
,
b
n
=
1
,
a
b
=
a
k
(
m
,
n
,
k
∈
Z
+
)
with
(
m
,
k
-
1
)
=
1
and
n
=
ind
m
(
k
)
,
the smallest positive integer for which
k
n
=
1
(
mod
m
)
,
with the conjugate of
a
by
b
written
a
b
=
b
-
1
a
b
.
The
right
and
left commutation semigroups of
G
, denoted
P
(
G
)
and
Λ
(
G
)
,
are the semigroups of mappings generated by
ρ
(
g
)
:
G
→
G
and
λ
(
g
)
:
G
→
G
defined by
(
x
)
ρ
(
g
)
=
[
x
,
g
]
and
(
x
)
λ
(
g
)
=
[
g
,
x
]
,
where the commutator of
g
and
h
is defined as
[
g
,
h
]
=
g
-
1
h
-
1
g
h
.
This paper builds on a previous study of commutation semigroups of dihedral groups conducted by the authors with C. Levy. Here we show that a similar approach can be applied to
G
, a metacyclic group with trivial centre. We give a construction of
P
(
G
)
and
Λ
(
G
)
as unions of
containers
, an idea presented in the previous paper on dihedral groups. In the case that
a
is cyclic of order
p
or
p
2
or its index is prime, we show that both
P
(
G
)
and
Λ
(
G
)
are disjoint unions of maximal containers. In these cases, we give an explicit representation of the elements of each commutation semigroup as well as formulas for their exact orders. Finally, we extend a result of J. Countryman to show that, for
G
(
m
,
n
,
k
) with
m
prime, the condition
P
(
G
)
=
Λ
(
G
)
is equivalent to
P
(
G
)
=
Λ
(
G
)
.</description><subject>Algebra</subject><subject>Commutation</subject><subject>Commutators</subject><subject>Containers</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Research Article</subject><subject>Unions</subject><issn>0037-1912</issn><issn>1432-2137</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp9kEFLAzEQhYMoWKt_wNOC5-hMZrdpjlKsCgUveg67abamdDc1ySr990a34M3T8GbeewMfY9cItwgg7yKAIOIggGetJKcTNsGSBBdI8pRNAEhyVCjO2UWMW8gaZjRhy4XvuiHVyfm-iLZzm-CHfSx8W7Sud8kWnU21OZidM8Xx9uXSe5GC-3T1rjC2T8FesrO23kV7dZxT9rZ8eF088dXL4_PifsWNkJC4tVVZW2xBteUchJREWIpGNjVWqrTtfC2bORhFYBVK02BeoWpoti6VRYM0ZTdj7z74j8HGpLd-CH1-qUUJFQlSosouMbpM8DEG2-p9cF0dDhpB__DSIy-deelfXppyiMZQzOZ-Y8Nf9T-pb3T2bZU</recordid><startdate>20200601</startdate><enddate>20200601</enddate><creator>DeWolf, Darien</creator><creator>Edmunds, Charles C.</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20200601</creationdate><title>Commutation semigroups of finite metacyclic groups with trivial centre</title><author>DeWolf, Darien ; Edmunds, Charles C.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-ee54ae1f09f48027733142b7ba1594ef8d7b80c930e917cb1ef819b36d49e1c13</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>Algebra</topic><topic>Commutation</topic><topic>Commutators</topic><topic>Containers</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Research Article</topic><topic>Unions</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>DeWolf, Darien</creatorcontrib><creatorcontrib>Edmunds, Charles C.</creatorcontrib><collection>CrossRef</collection><jtitle>Semigroup forum</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>DeWolf, Darien</au><au>Edmunds, Charles C.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Commutation semigroups of finite metacyclic groups with trivial centre</atitle><jtitle>Semigroup forum</jtitle><stitle>Semigroup Forum</stitle><date>2020-06-01</date><risdate>2020</risdate><volume>100</volume><issue>3</issue><spage>765</spage><epage>789</epage><pages>765-789</pages><issn>0037-1912</issn><eissn>1432-2137</eissn><abstract>We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented
G
(
m
,
n
,
k
)
=
a
,
b
;
a
m
=
1
,
b
n
=
1
,
a
b
=
a
k
(
m
,
n
,
k
∈
Z
+
)
with
(
m
,
k
-
1
)
=
1
and
n
=
ind
m
(
k
)
,
the smallest positive integer for which
k
n
=
1
(
mod
m
)
,
with the conjugate of
a
by
b
written
a
b
=
b
-
1
a
b
.
The
right
and
left commutation semigroups of
G
, denoted
P
(
G
)
and
Λ
(
G
)
,
are the semigroups of mappings generated by
ρ
(
g
)
:
G
→
G
and
λ
(
g
)
:
G
→
G
defined by
(
x
)
ρ
(
g
)
=
[
x
,
g
]
and
(
x
)
λ
(
g
)
=
[
g
,
x
]
,
where the commutator of
g
and
h
is defined as
[
g
,
h
]
=
g
-
1
h
-
1
g
h
.
This paper builds on a previous study of commutation semigroups of dihedral groups conducted by the authors with C. Levy. Here we show that a similar approach can be applied to
G
, a metacyclic group with trivial centre. We give a construction of
P
(
G
)
and
Λ
(
G
)
as unions of
containers
, an idea presented in the previous paper on dihedral groups. In the case that
a
is cyclic of order
p
or
p
2
or its index is prime, we show that both
P
(
G
)
and
Λ
(
G
)
are disjoint unions of maximal containers. In these cases, we give an explicit representation of the elements of each commutation semigroup as well as formulas for their exact orders. Finally, we extend a result of J. Countryman to show that, for
G
(
m
,
n
,
k
) with
m
prime, the condition
P
(
G
)
=
Λ
(
G
)
is equivalent to
P
(
G
)
=
Λ
(
G
)
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00233-020-10097-3</doi><tpages>25</tpages></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0037-1912 |
| ispartof | Semigroup forum, 2020-06, Vol.100 (3), p.765-789 |
| issn | 0037-1912 1432-2137 |
| language | eng |
| recordid | cdi_proquest_journals_2405323925 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Algebra Commutation Commutators Containers Mathematics Mathematics and Statistics Research Article Unions |
| title | Commutation semigroups of finite metacyclic groups with trivial centre |
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