mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 ...

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

of the

cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

''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 grou ...

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.

Representation theory

There is a natural representation of elements of

S(m,n)

as generalized permutation matrices, where the nonzero entries are ''m''-th

roots of unity In mathematics, a root of unity 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 number theory, the theory of group char ...

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 modules; see .

Homology

The first group homology 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 s ...

– is

Z_m \times Z_2

(for ''m'' odd this is

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

Z_

): the

Z_m

factors (which are all conjugate, hence must map identically in an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

, 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 \ope ...

– 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