Exercise 983Algebra › GroupsL1Added 3 Nov 2026

The symmetric group $S_3$

#symmetric group#permutations#order

Statement

Compute the order of $S_3$ , list its elements, and identify the subgroup of even permutations.

Hint

$|S_n|=n!$ . Permutations of $\{1,2,3\}$ : $6$ in total, three even and three odd.

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Solution

Order. $|S_3|=3!=6$ .

Elements.
- Identity: $e$ .
- Transpositions (odd): $(12),(13),(23)$ .
- $3$ -cycles (even): $(123),(132)$ .

Even permutations. $\{e,(123),(132)\}$ , which forms a subgroup of order $3$ isomorphic to $\mathbb{Z}/3\mathbb{Z}$ (the alternating group $A_3$ ).

Conclusion. $|S_3|=6$ , $A_3$ has order $3$ and is normal (index $2$ ).

The symmetric group $S_3$

Statement

Solution

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