Permutation group

In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). Note that the group of all permutations of a set is the symmetric group; the term permutation group is usually restricted to mean a subgroup of the symmetric group. The symmetric group of n elements is denoted by Sn; if M is any finite or infinite set, then the group of all permutations of M is often written as Sym(M).

Isomorphisms

If G and H are two permutation groups on the same set S, then we say that G and H are isomorphic as permutation groups if there exists a bijective map f : S → S such that r |-> f −1 o r o f defines a bijective map between G and H; in other words, if for each element g in G, there is a unique hg in H such that for all s in S, (g o f)(s) = (f o hg)(s). In this case, G and H are also isomorphic as groups.

Related Topics:
Isomorphic - Isomorphic as groups

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Notice that different permutation groups may well be isomorphic as abstract groups, but not as permutation groups. For instance, the permutation group on {1,2,3,4} described above is isomorphic as a group (but not as a permutation group) to {(1)(2)(3)(4), (12)(34), (13)(24), (14)(23)}. Both are isomorphic as groups to the Klein group V4.

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If (G,M) and (H,M) such that both G and H are isomorphic as groups to Sym(M), then (G,M) and (H,M) are isomorphic as permutation groups; thus it is appropriate to talk about the symmetric group Sym(M) (up to isomorphism).

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