Signed symmetric group: Difference between revisions

Revision as of 22:11, 7 April 2010

Definition

The signed symmetric group or hyperoctahedral group of degree $n$ is defined in the following equivalent ways:

It is the external wreath product of the cyclic group of order two $Z_{2}$ with the symmetric group of degree $n$ with its natural action on a set of size $n$ . In symbols, it is $Z_{2} ≀ S_{n}$ .
It is the external semidirect product of an elementary abelian group of order $2^{n}$ and a symmetric group of degree $n$ , acting as coordinate permutations in the natural way.
It is the subgroup of the general linear group over integers $G L (n, Z)$ 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 $\pm 1$ . 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 $R^{n}$ .
It is the centralizer in the symmetric group of degree $2 n$ of a permutation that is a product of $n$ disjoint transpositions.

Arithmetic functions

All these function values are

Function	Value	Explanation
order	$2^{n} \cdot n!$
exponent	lcm of $1, 2, \dots, n$ (?)

Particular cases

Value of $n$	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 WreathProduct, [{GAP:CyclicGroup|CyclicGroup]], and SymmetricGroup. For a given $n$ , the signed symmetric group is given by:

WreathProduct(CyclicGroup(2),SymmetricGroup(n))

For instance, for $n = 4$ , the group can be defined as:

WreathProduct(CyclicGroup(2),SymmetricGroup(4))

@@ Line 5: / Line 5: @@
 # 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 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 centralizer in the symmetric group of degree <math>2n</math> of a permutation that is a product of <math>n</math> disjoint transpositions.