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How do you prove Cayley's theorem which states that every group is isomorphic to a permutation group?
Cayley's theorem:Let (G,$) be a group. For each g Є G, let Jg be a permutation of G such that
Jg(x) = g$x
J, then, is a function from g to Jg, J: g --> Jg and is an isomorphism from (G,$) onto a permutation group on G.
Proof:We already know, from another established theorem that I'm not going to prove here, that an element invertible for an associative composition is cancellable for that composition, therefore Jg is a permutation of G. Given another permutation, Jh = Jg, then h = h$x = Jh(x) = Jg(x) = g$x = g, meaning J is injective.Now for the fun part!
For every x Є G, a composition of two permutations is as follows:
(Jg â—‹ Jh)(x) = Jg(Jh(x)) = Jg(h$x) = g$(h$x) = (g$h)$x = Jg$h(x)
Therefore Jg ○ Jh = Jg$h(x) for all g, h Є G
That means that the set Ђ = {Jg: g Є G} is a stable subset of the permutation subset of G, written as ЖG, and J is an isomorphism from G onto Ђ. Consequently, Ђ is a group and therefore is a permutation group on G.
Q.E.D.
What else can I help you with?
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What is Cayley's Theorem?
Cayley's Theorem states that every group G is isomorphic to a subgroup of the symmetric group on G.
What this theorem states?
It states NOTHING!
What states the Pythagorean theorem?
Pythagoras' theorem states that for any right angle triangle that the square of its hypotenuse is equal to the sum of its squared sides.
How do you count spanning trees in a graph?
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pythagorean theorem.
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the Pythagorean Theorem
Which theorem states that the measure of an exterior angle in a triangle is the sum of its remote interior angle measures?
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