# Group Action on a Set

#### 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.

Related Linear and Abstract Algebra News on Phys.org

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}.

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving