# Group Action on a Set

1. Dec 1, 2008

### Symmetryholic

Let g= $$\left( \begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \\ 2 & 5 & 4 & 1 & 3 \end{array} \right)$$ be an element of $$S_{5}$$ and a set S={1,2,3}.

The theorem of a group action says "If a group G acts on a set, this action induces a homomorphism G->A(S), A(S) is the group of all permutations of the set S."

When I apply the above action g on a set S, $$1 \mapsto 2, 2 \mapsto 5, 3 \mapsto 4$$, which is not a permutation of a set S.

A group action on a set possibly does not induce a set of its own permutation on set S?

Any opinion will be appreciated.

2. Dec 2, 2008