odd order

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

odd order

Postby brain » Mon Dec 20, 2010 9:26 am

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

Postby logaritym » Mon Dec 20, 2010 9:37 am

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

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

Re: odd order

Postby bobn » Mon Dec 20, 2010 9:51 am

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

Postby brain » Mon Dec 20, 2010 11:22 am

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

Postby bobn » Mon Dec 20, 2010 12:36 pm

Yup, provided o(Group) > 1

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

Re: odd order

Postby enork » Tue Dec 21, 2010 12:27 am

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




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