Question

Let G be a group and let g,h in G with h(g^2)h = (h^2)gh. Show that g = h.

Answer 1

@TedG This takes some cancellation laws... you up for it? :)

Answer 2

yh im up for it.

Answer 3

Okay, so let's remove all the fancy exponents and put everything at face value... we know that...
\[\huge hg^2h=h^2gh\\ \huge hggh = hhgh\]

right?

Answer 4

yh makes sense

Answer 5

Now, since this is a group, the inverses of g and h would exist, right?
\[\huge g^{-1} \ , \ h^{-1} \ \in G\]

Answer 6

of course.

Answer 7

Now, we know this...
\[\large hggh = hhgh\]so we can (sneakily) multiply both sides of the equation by \(\large h^{-1}\) on the right

\[\large hgghh^{-1}=hhghh^{-1}\]
See where I'm going with this?

Answer 8

yh i see

Answer 9

so we can then cancel the g, end up with hg=hh

Answer 10

yup... and then...?
We can do the same trick with the \(\large h^{-1}\) multiplying on the left, this time...

Answer 11

oh so you can just multiply on the left, which i guess makes sense as the inverse is two sided.

Answer 12

Yup... leaving you with...? ;)

Answer 13

g=h god i didnt think to just lay it out as a simple equation and use the inverses, i was over complicating it using that 1= gg^-1

Answer 14

There there :)

Answer 15

Thank you for that, Much appreciated.

Answer 16

You major in Maths?

Answer 17

i am studying in uk, not sure the definition of major but yes i am taking a degree in maths