# Symmetric groups

Given the set Sn of the permutations of the n elements of a set S, the set Sn forms a group (Sn, o) with respect to the permutation operation o.

$$(S_n, \text{o})$$

The group of all permutations Sn of the n symbols is called a symmetric group.

The subgroups of Sn are called permutation groups.

Note. Permutation groups and symmetric groups are non-Abelian (non-commutative) groups because the permutation operation o is non-commutative.

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Group theory

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