Redirected from Cayley's theorem
A permutation of a set G is any bijective function taking G onto G; and the set of all such functions forms a group under function composition, called the symmetric group on G, and written as Sym(G).
Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (R,+)) as a permutation group of some underlying set. Thus, theorems which are true for permutation groups are true for groups in general.
From elementary group theory, we can see that for any element g in G, we must have g*G = G; and by cancellation rules, that g*x = g*y if and only if x = y. So multiplication by g acts as a bijective function f_{g} : G → G, by defining f_{g}(x) = g*x. Thus, f_{g} is a permutation of G, and so is a member of Sym(G).
The subset K of Sym(G) defined as K = {f_{g} : g in G and f_{g}(x) = g*x for all x in G} is a subgroup of Sym(G) which is isomorphic to G. The fastest way to establish this is to consider the function T : G → Sym(G) with T(g) = f_{g} for every g in G. T is a group homomorphism because (using "•" for composition in Sym(G)):
Thus G is isomorphic to the image of T, which is the subgroup K considered earlier.
T is sometimes called the regular representation of G.
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