Open main menu

Groupprops β

Signed symmetric group

Definition

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

  1. It is the external wreath product of the cyclic group of order two \mathbb{Z}_2 with the symmetric group of degree n with its natural action on a set of size n. In symbols, it is \mathbb{Z}_2 \wr S_n.
  2. 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.
  3. It is the subgroup of the general linear group over integers GL(n,\mathbb{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.
  4. It is the group of symmetries of the hyperoctahedron in \mathbb{R}^n.
  5. It is the centralizer in the symmetric group of degree 2n of a permutation that is a product of n disjoint transpositions.
  6. It is the generalized symmetric group S(2,n).
  7. It is a Coxeter group with a particular kind of Coxeter presentation (for more, see signed symmetric group of finite degree is a Coxeter group)

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 = 2^n \cdot n! 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, 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))