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If a is even, prove a^(1) is even...by Zondrina
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#1
Nov1212, 10:10 PM

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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 2cycles. 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? 


#2
Nov1212, 10:35 PM

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#3
Nov1212, 11:19 PM

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#4
Nov1312, 09:14 AM

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If a is even, prove a^(1) is even...



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