(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If a is even, prove a^{-1}is even.

2. Relevant equations

We know that every permutation in [itex]S_n, n>1[/itex] can be written as a product of 2-cycles. Also note that the identity can be expressed as (12)(12) for this to be possible.

3. The attempt at a solution

Suppose a is a permutation made up of 2cycles, say [itex]a_1, ...,a_n[/itex].

We know that :

[itex]a^{-1} = (a_1, ...,a_n)^{-1} = a_{1}^{-1}, ..., a_{n}^{-1}[/itex]

Now since we can write (ab) = (ba) for any two cycle, we know : [itex]a^{-1} = (a_1, ...,a_n)^{-1} = a_{1}^{-1}, ..., a_{n}^{-1} = a_1, ...,a_n = a[/itex]

So if a is an even permutation, it means that |a| is even, say |a|=n. Then |a^{-1}| is also even since |a| = |a^{-1}| for 2cycles.

Thus if a is even, then a^{-1}is also even.

Is this correct?

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# Homework Help: If a is even, prove a^(-1) is even

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