Clarification of Permuatation Question

[Redhead711]

NEED HELP! with permutation proof

Sorry for any confusion the question I have is:

Show that a permutation with odd order must be an even permutation.

The order of a permutation of a finite set written in disjoint cycle form
is the least common multiple of the lengths of the cycles.

This is what I have worked out so far:

Let e = epsilon
Let B = a permutation
Let k = any integer

Now say B^(2k+1) = e. Where B^(2k+1) is an odd permutation.
Then B^(2k)= B^(-1).
But B^(2k) = B^(k)^2 is even.

I would really appreciate some help in putting this
proof together in a more coherent fashion.
I am very confused. Thanks

 

[matt grime]

More important than that surely is the definition of the sign (even or odd) at no point do you use its properties so surely something must be telling you you need some more details.

sign is multiplicative: sign(xy)=sign(x)sign(y)

so it suffices to show that an element of odd order cannot possesses any cycles of odd sign (when written as of disjoint cycles), which is where you're order being the lcm comes in

 

[Redhead711]

Thanks to Matt for all your help. I understand the problem better know and have been able to solve it. I'm very grateful