Generalized symmetric group

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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 ...
$S\left(m,n\right) := Z_m \wr S_n$ 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 binary ...
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 \m ...
of order ''n''.

Examples

* For $m=1,$ the generalized symmetric group is exactly the ordinary symmetric group: $S\left(1,n\right) = S_n.$ * 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 $S\left(2,n\right)$ with the signed symmetric group.

Representation theory

There is a natural representation of elements of $S\left(m,n\right)$ as generalized permutation matrices, 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 ...
: $Z_m \cong \mu_m.$ 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 ...
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 l ...
group (concretely, the abelianization) 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) is given by : :$H_2\left(S\left(2k+1,n\right)\right) = \begin 1 & n < 4\\ \mathbf/2 & n \geq 4.\end$ :$H_2\left(S\left(2k+2,n\right)\right) = \begin 1 & n = 0, 1\\ \mathbf/2 & n = 2\\ \left(\mathbf/2\right)^2 & n = 3\\ \left(\mathbf/2\right)^3 & n \geq 4. \end$ Note that it depends on ''n'' and the parity of ''m:'' $H_2\left(S\left(2k+1,n\right)\right) \approx H_2\left(S\left(1,n\right)\right)$ and $H_2\left(S\left(2k+2,n\right)\right) \approx H_2\left(S\left(2,n\right)\right),$ which are the Schur multipliers of the symmetric group and signed symmetric group.

References

* * * {{refend Permutation groups