Associative

We know

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

We will Let

x = (1 2)(3)(4)(5)(6)

which means that 1 is replaced with 2 and 2 is replaced with 1

and since the only change in the permutation is (1 2) we can just write

x = (1 2)

We will let

y = (1)(2)(3)(4)(5 6)

which can be written simply as

y = (5 6)

Then we can write xy as

xy = (1 2)(5 6)

similarly

yx = (5 6)(1 2)

which can be written as (1 2)(5 6)

Therefore xy=yx and S6 is associative.

Also see

Identity

Inverse

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