Encyclopedia > Cayley's theorem

  Article Content

Cayley's Theorem

Redirected from Cayley's theorem

In group theory, Cayley[?]'s Theorem states that every group is isomorphic to a subgroup of the symmetric group on G. This can be understood as an example of the group action of G on the elements of G.

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.

Proof of the theorem

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 fg : GG, by defining fg(x) = g*x. Thus, fg is a permutation of G, and so is a member of Sym(G).

The subset K of Sym(G) defined as K = {fg : g in G and fg(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) = fg for every g in G. T is a group homomorphism because (using "•" for composition in Sym(G)):

(fgfh)(x) = fg(fh(x)) = fg(h*x) = g*(h*x) = (g*h)*x = f(g*h)(x),
for all x in G, and hence
T(g) • T(h) = fgfh = f(g*h) = T(g*h).
The homomorphism T is also injective since T(g) = idG (the identity element of Sym(G)) implies that g*x = x for all x in G, and taking x to be the identity element e of G yields g = g*e = e.

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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Islandia, New York

... up of individuals and 2.3% have someone living alone who is 65 years of age or older. The average household size is 3.04 and the average family size is 3.49. In the ...

 
 
 
This page was created in 22.2 ms