In mathematics, the generalized symmetric group is the

wreath product In group theory, the wreath product is a special combination of two groups based on the semidirect product. It is formed by the action of one group on many copies of another group, somewhat analogous to exponentiation. Wreath products are used i ...

S(m,n) := Z_m \wr S_n

of the

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

of order ''m'' and the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

of order ''n''.

Examples

* For

m=1,

the generalized symmetric group is exactly the ordinary symmetric group:

S(1,n) = S_n.

* For

m=2,

one can consider the cyclic group of order 2 as positives and negatives (

Z_2 \cong \

) and identify the generalized symmetric group

S(2,n)

with the

signed symmetric group In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a parame ...

Representation theory

There is a natural representation of elements of

S(m,n)

generalized permutation matrices In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the ...

, where the nonzero entries are ''m''-th

roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...

Z_m \cong \mu_m.

The representation theory has been studied since ; see references in . As with the symmetric group, the representations can be constructed in terms of

Specht module In mathematics, a Specht module is one of the representations of symmetric groups studied by . They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of ''n'' form a complete set of irreducible representations of t ...

s; see .

Homology

The first

group homology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology lo ...

group (concretely, the

abelianization In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal ...

) is

Z_m \times Z_2

(for ''m'' odd this is isomorphic to

Z_

): the

Z_m

factors (which are all conjugate, hence must map identically in an abelian group, since conjugation is trivial in an abelian group) can be mapped to

Z_m

(concretely, by taking the product of all the

Z_m

values), while the sign map on the symmetric group yields the

Z_2.

These are independent, and generate the group, hence are the abelianization. The second homology group (in classical terms, the

Schur multiplier In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \oper ...

) is given by : :

H_2(S(2k+1,n)) = \begin 1 & n < 4\\
\mathbf/2 & n \geq 4.\end

H_2(S(2k+2,n)) = \begin 1 & n = 0, 1\\
\mathbf/2 & n = 2\\
(\mathbf/2)^2 & n = 3\\
(\mathbf/2)^3 & n \geq 4.
\end

Note that it depends on ''n'' and the parity of ''m:''

H_2(S(2k+1,n)) \approx H_2(S(1,n))

and

H_2(S(2k+2,n)) \approx H_2(S(2,n)),

which are the Schur multipliers of the symmetric group and signed symmetric group.

References

* * * {{refend Permutation groups