Group theory MOC

Symmetric group

A symmetric group S𝑛 of degree 𝑛 is a group of order 𝑛! made up of permutations of 𝑛 objects. Let ℕ𝑛 ={1,…,𝑛}. Then S𝑛 is the set of all bijections ℕ𝑛 →ℕ𝑛, i.e. \mathrm{S}_{n} = \mathrm{Aut}_{\Set}(\mathbb{N}_{n}) = \mathbb{N}_{n}! group

Each permutation in a symmetric group may be written as a product of disjoint 𝑛 cycles, which is unique up to order of cycles and 1-cycles may be added or dropped.¹

(1234553421)=(15)(234)

In a sense, symmetry groups are the largest (by order) possible groups with a given structure, as formalised by Cayley’s theorem – Every group is a subgroup of a symmetry group.

Properties

• Alternating character
• Conjugacy classes of a symmetric group are determined by cycle structure
• Normal subgroups of the symmetric group
  • Alternating group

See also

• Representation theory of finite symmetric groups

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Footnotes

1. 1996, Representations of finite and compact groups, §I.3, p. 9 ↩