Inverse

We will let

x = (1 2)(3)(4)(5)(6)

which can be written as

x = (1 2)

Since replaces 1 with 2 and 2 with 1 we know that if we perform x again we will get our original permutation

xx = (1)(2)(3)(4)(5)(6)

because 2 is replaced with 1 and 1 is replaced with 2. This recreates our our original permutation.

Therefore

e == xx == (1)(2)(3)(4)(5)(6)

This shows that x has an inverse which is x itself.

This now shows that S6 has an inverse

Also see

Identity

Associative

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