Group action?

Question

G simple group of order 60.Show if X is a set with elements < 4,then the only group action of G on X is the trivial action

Answer 1

Let X have some finite number n of elements.

Let Sym(X) be the full permutation group on X, with n! elements.

Each g in G acts on X by permuting its elements. Let f : G -> Sym(X) be given by f(g) is the map a -> ag for all a in X.

f is a homomorphism since (a g1) g2 = a (g1 g2).

Let K be the kernel of f. K is a normal subgroup of G and since G is simple, K must either be the one element subgroup or all of G.

If K is the one element subgroup, then f is 1-1 and so Sym(X) = n! >= |G| = 60.

But for n = 1, 2, 3, 4, you have n! < 60.

So G is simple, X has n elements, and n! < |G|, then the only group action of g on X is the trivial action.

Answer 2

~If Godzilla and King Kong gang up on Mothra, the Green Lantern is in a world of kaka.