Inverse of Group Elements: Find g_i^-1g_j^-1

[LagrangeEuler]

Homework Statement

Find $(g_ig_j)^{-1}$ for any two elements of group $G$.

Homework Equations

For matrices $(AB)^{-1}=B^{-1}A^{-1}$

The Attempt at a Solution

I'm not sure how to show this? I could show that for matrices $(AB)^{-1}=B^{-1}A^{-1}$. And that for numbers
$(ab)^{-1}=a^{-1}b^{-1}$

 

[tiny-tim]

Hi LagrangeEuler!

Find $(g_ig_j)^{-1}$ for any two elements of group $G$.

If h = $(g_ig_j)^{-1}$, then $hg_ig_j$ = I.

Sooo what combinations of gs would h have to be made of? ​

 

[LagrangeEuler]

Could I just write from that relation that $(g_ig_j)^{-1}=g_j^{-1}g_i^{-1}$? :)
It looks obvious but what is right mathematical way to write it? :)

 

[jbunniii]

Could I just write from that relation that $(g_ig_j)^{-1}=g_j^{-1}g_i^{-1}$? :)
It looks obvious but what is right mathematical way to write it? :)

Check whether it satisfies the properties of the inverse. If $h$ is the inverse of $g$, then $hg = gh = 1$. See if your $h$ satisfies this for $g = g_i g_j$.