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Uni Stuff => Science => Faculties => Mathematics => Topic started by: kamil9876 on October 07, 2009, 07:28:23 pm

Title: Groups
Post by: kamil9876 on October 07, 2009, 07:28:23 pm: Let $G$ be a group with only one element of order 2, say $g$ . Prove that $gx=xg$ for all $x \in G$
Title: Re: Groups
Post by: kamil9876 on October 07, 2009, 10:36:42 pm: finally solved it :) Want to see if there are more general solutions/better than mine/ remarks on general theory.

After some playing around:

I noticed that:

let x be any element in G:
$<br />xgx^{-1} \in G$ by closure.
$<br />(xgx^{-1})^2=xgx^{-1}xgx^{-1}=xggx^{-1}=xx^{-1}=1$

Because g is the only element of order 2. $y^2=1$ has only two solutions: 1 and g.

But because, $(xgx^{-1})^2=1$ for all $x \in G$ then $xgx^{-1}=y$ , namely $xgx^{-1}=g or 1$

First case:

$xgx^{-1}=g$
$<br />\implies xg=gx$ (1)

While the second case is impossible since it implies:

$xg=x$
$\implies g=1$ but g has order 2 so this cannot be the case. Hence (1) is the only case.