Group Action on Set S: The Induced Homomorphism from g \in S_{5}

[Symmetryholic]

Let g= $$\left( \begin{array}{ccccc} <br /> <br /> 1 & 2 & 3 & 4 & 5 \\ <br /> <br /> 2 & 5 & 4 & 1 & 3 \end{array} \right) <br />$$ 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.

 

[adriank]

The problem is that S₅ doesn't act on {1, 2, 3} (with the usual action of permutation groups), since 2 is in X, but g(2) = 5 is not in {1, 2, 3}. A group action of G on a set X must send an element of G and an element of X to an element of X; what you show is a map that does not map into {1, 2, 3}, so it cannot be an action on {1, 2, 3}.