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odd order

Posted: Mon Dec 20, 2010 9:26 am
by brain
In a group G let x be of odd order. It is true that there is y in G so that $$y^2=x$$?

Re: odd order

Posted: Mon Dec 20, 2010 9:37 am
by logaritym
x^{2n+1}=e
Then
(x^{n+1})^2=x^{2n+2}=x.
y^2=x, where y=x^{n+1}.
Answer: Yes.

Re: odd order

Posted: Mon Dec 20, 2010 9:51 am
by bobn
Say S = { e, x, x^2,..............x^n-1 } be a subgroup, where x^n = e and n is odd.

say y = x ^ (n+1)/2 ; y^2 = x^(n+1) = x

Re: odd order

Posted: Mon Dec 20, 2010 11:22 am
by brain
So we can also say that there is y such that $$x^2=y$$, right?

Re: odd order

Posted: Mon Dec 20, 2010 12:36 pm
by bobn
Yup, provided o(Group) > 1

Re: odd order

Posted: Tue Dec 21, 2010 12:27 am
by enork
The trivial group is fine. e^2 = e.