Prove the right and left cancellation laws for a group G tha

Prove the right and left cancellation laws for a group G; that is, show that in the group G, ba = ca implies b = c and ab = ac implies b = c for elements a, b, c element G. Show that if a^2 = e for all elements a in a group G, then G must be abelian.

Solution

31 ) Take two elements g,h   which belong to G.

We know that (gh)^2 = e

and so you get

gh*gh = e.

Applying h to both sides on the right gives you

ghg hh = e.h = hg

Similarly , applying g to both sides :

gh gg = hg

gh = hg

Proving G is abelian