I need to prove the following:
There exists an x in G s.t. xax=b <=> c<sup>2</sup>=ab for some c.
I have shown the \(\displaystyle =>\) and have found that c=x<sup>-1</sup>b.
xax=b <=>
xa=bx<sup>-1</sup> <=>
xab=bx<sup>-1</sup>b <=>
ab=x<sup>-1</sup>bx<sup>-1</sup>b = (x<sup>-1</sup>b)<sup>2</sup>
Is that right?
But I am having trouble showing the <= direction.
There exists an x in G s.t. xax=b <=> c<sup>2</sup>=ab for some c.
I have shown the \(\displaystyle =>\) and have found that c=x<sup>-1</sup>b.
xax=b <=>
xa=bx<sup>-1</sup> <=>
xab=bx<sup>-1</sup>b <=>
ab=x<sup>-1</sup>bx<sup>-1</sup>b = (x<sup>-1</sup>b)<sup>2</sup>
Is that right?
But I am having trouble showing the <= direction.