## odd order

Forum for the GRE subject test in mathematics.
brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

### odd order

In a group G let x be of odd order. It is true that there is y in G so that $y^2=x$?

logaritym
Posts: 42
Joined: Sun Nov 14, 2010 10:06 am

### Re: odd order

x^{2n+1}=e
Then
(x^{n+1})^2=x^{2n+2}=x.
y^2=x, where y=x^{n+1}.

bobn
Posts: 61
Joined: Thu Nov 12, 2009 2:59 am

### Re: odd order

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

brain
Posts: 28
Joined: Tue Aug 25, 2009 12:16 pm

### Re: odd order

So we can also say that there is y such that $x^2=y$, right?

bobn
Posts: 61
Joined: Thu Nov 12, 2009 2:59 am

### Re: odd order

Yup, provided o(Group) > 1

enork
Posts: 33
Joined: Fri Sep 18, 2009 3:16 am

### Re: odd order

The trivial group is fine. e^2 = e.