Signed symmetric group - Revision history

Vipul: /* Particular cases */

2011-09-14T21:25:05Z

Particular cases

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	==Particular cases==		==Particular cases==

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	! Value of <math>n</math> !! Order !! GAP ID !! Common name		! Value of <math>n</math> !! Order = <math>2^n \cdot n!</math>!! GAP ID !! Common name
	\|-		\|-
	\| 0 \|\| 1 \|\| (1,1) \|\| [[trivial group]]		\| 0 \|\| 1 \|\| (1,1) \|\| [[trivial group]]

Vipul: /* Definition */

2011-03-30T20:49:04Z

Definition

← Older revision		Revision as of 20:49, 30 March 2011
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	# It is the centralizer in the symmetric group of degree <math>2n</math> of a permutation that is a product of <math>n</math> disjoint transpositions.		# It is the centralizer in the symmetric group of degree <math>2n</math> of a permutation that is a product of <math>n</math> disjoint transpositions.
	# It is the [[defining ingredient::generalized symmetric group]] <math>S(2,n)</math>.		# It is the [[defining ingredient::generalized symmetric group]] <math>S(2,n)</math>.
			# It is a [[defining ingredient::Coxeter group]] with a particular kind of Coxeter presentation (for more, see [[signed symmetric group of finite degree is a Coxeter group]])

	==Arithmetic functions==		==Arithmetic functions==

Vipul: /* GAP implementation */

2010-11-02T02:58:22Z

GAP implementation

← Older revision		Revision as of 02:58, 2 November 2010
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	==GAP implementation==		==GAP implementation==

	The groups can be constructed using GAP, with the help of the functions [[GAP:WreathProduct\|WreathProduct]], [{GAP:CyclicGroup\|CyclicGroup]], and [[GAP:SymmetricGroup\|SymmetricGroup]]. For a given <math>n</math>, the signed symmetric group is given by:		The groups can be constructed using GAP, with the help of the functions [[GAP:WreathProduct\|WreathProduct]], [[GAP:CyclicGroup\|CyclicGroup]], and [[GAP:SymmetricGroup\|SymmetricGroup]]. For a given <math>n</math>, the signed symmetric group is given by:

	<tt>WreathProduct(CyclicGroup(2),SymmetricGroup(n))</tt>		<tt>WreathProduct(CyclicGroup(2),SymmetricGroup(n))</tt>

Vipul at 22:22, 7 April 2010

2010-04-07T22:22:38Z

← Older revision		Revision as of 22:22, 7 April 2010
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			{{variation of\|symmetric group}}
	==Definition==		==Definition==

Vipul at 22:19, 7 April 2010

2010-04-07T22:19:03Z

← Older revision		Revision as of 22:19, 7 April 2010
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	# It is the group of symmetries of the hyperoctahedron in <math>\mathbb{R}^n</math>.		# It is the group of symmetries of the hyperoctahedron in <math>\mathbb{R}^n</math>.
	# It is the centralizer in the symmetric group of degree <math>2n</math> of a permutation that is a product of <math>n</math> disjoint transpositions.		# It is the centralizer in the symmetric group of degree <math>2n</math> of a permutation that is a product of <math>n</math> disjoint transpositions.
			# It is the [[defining ingredient::generalized symmetric group]] <math>S(2,n)</math>.

	==Arithmetic functions==		==Arithmetic functions==

Vipul at 22:11, 7 April 2010

2010-04-07T22:11:01Z

← Older revision		Revision as of 22:11, 7 April 2010
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	# It is the [[defining ingredient::external wreath product]] of [[defining ingredient::cyclic group:Z2\|the cyclic group of order two]] <math>\mathbb{Z}_2</math> with the [[symmetric group]] of degree <math>n</math> with its natural action on a set of size <math>n</math>. In symbols, it is <math>\mathbb{Z}_2 \wr S_n</math>.		# It is the [[defining ingredient::external wreath product]] of [[defining ingredient::cyclic group:Z2\|the cyclic group of order two]] <math>\mathbb{Z}_2</math> with the [[symmetric group]] of degree <math>n</math> with its natural action on a set of size <math>n</math>. In symbols, it is <math>\mathbb{Z}_2 \wr S_n</math>.
	# It is the [[defining ingredient::external semidirect product]] of an [[elementary abelian group]] of order <math>2^n</math> and a [[symmetric group]] of degree <math>n</math>, acting as coordinate permutations in the natural way.		# It is the [[defining ingredient::external semidirect product]] of an [[elementary abelian group]] of order <math>2^n</math> and a [[symmetric group]] of degree <math>n</math>, acting as coordinate permutations in the natural way.
	# It is the subgroup of the [[general linear group over integers]] <math>GL(n,\mathbb{Z})</math> comprising all matrices which have exactly one nonzero entry in each row and exactly one nonzero entry in each column, and the nonzero entries are all <math>\pm 1</math>.		# It is the subgroup of the [[general linear group over integers]] <math>GL(n,\mathbb{Z})</math> comprising all matrices which have exactly one nonzero entry in each row and exactly one nonzero entry in each column, and the nonzero entries are all <math>\pm 1</math>. More generally, it can be realized using these matrices over any ring of characteristic not equal to two.
	# It is the group of symmetries of the hyperoctahedron in <math>\mathbb{R}^n</math>.		# It is the group of symmetries of the hyperoctahedron in <math>\mathbb{R}^n</math>.
	# It is the centralizer in the symmetric group of degree <math>2n</math> of a permutation that is a product of <math>n</math> disjoint transpositions.		# It is the centralizer in the symmetric group of degree <math>2n</math> of a permutation that is a product of <math>n</math> disjoint transpositions.

Vipul: Created page with '==Definition== The '''signed symmetric group''' or '''hyperoctahedral group''' of degree $n$ is defined in the following equivalent ways: # It is the [[defining ingr…'

2010-04-07T22:08:35Z

Created page with '==Definition== The '''signed symmetric group''' or '''hyperoctahedral group''' of degree <math>n</math> is defined in the following equivalent ways: # It is the [[defining ingr…'

New page

==Definition==

The '''signed symmetric group''' or '''hyperoctahedral group''' of degree <math>n</math> is defined in the following equivalent ways:

# It is the [[defining ingredient::external wreath product]] of [[defining ingredient::cyclic group:Z2|the cyclic group of order two]] <math>\mathbb{Z}_2</math> with the [[symmetric group]] of degree <math>n</math> with its natural action on a set of size <math>n</math>. In symbols, it is <math>\mathbb{Z}_2 \wr S_n</math>.
# It is the [[defining ingredient::external semidirect product]] of an [[elementary abelian group]] of order <math>2^n</math> and a [[symmetric group]] of degree <math>n</math>, acting as coordinate permutations in the natural way.
# It is the subgroup of the [[general linear group over integers]] <math>GL(n,\mathbb{Z})</math> comprising all matrices which have exactly one nonzero entry in each row and exactly one nonzero entry in each column, and the nonzero entries are all <math>\pm 1</math>.
# It is the group of symmetries of the hyperoctahedron in <math>\mathbb{R}^n</math>.
# It is the centralizer in the symmetric group of degree <math>2n</math> of a permutation that is a product of <math>n</math> disjoint transpositions.

==Arithmetic functions==

All these function values are
{| class="wikitable" border="1"
! Function !! Value !! Explanation
|-
| [[order of a group|order]] || <math>2^n \cdot n! </math> ||
|-
| [[exponent of a group|exponent]] || lcm of <math>1,2,\dots,n</math> (?) ||
|}

==Particular cases==

{| class="wikitable" border="1"
! Value of <math>n</math> !! Order !! GAP ID !! Common name
|-
| 0 || 1 || (1,1) || [[trivial group]]
|-
| 1 || 2 || (2,1) || [[cyclic group:Z2]]
|-
| 2 || 8 || (8,3) || [[dihedral group:D8]]
|-
| 3 || 48 || (48,48) || [[direct product of S4 and Z2]]
|-
| 4 || 384 || (384,5602) ||
|}

==GAP implementation==

The groups can be constructed using GAP, with the help of the functions [[GAP:WreathProduct|WreathProduct]], [{GAP:CyclicGroup|CyclicGroup]], and [[GAP:SymmetricGroup|SymmetricGroup]]. For a given <math>n</math>, the signed symmetric group is given by:

<tt>WreathProduct(CyclicGroup(2),SymmetricGroup(n))</tt>

For instance, for <math>n = 4</math>, the group can be defined as:

<tt>WreathProduct(CyclicGroup(2),SymmetricGroup(4))</tt>