Let (G,*) be a group. Define a new binary operation • on G by setting a • b = b * a for all a,b in G. Prove that (G,•) is a group
Well, how is being a group defined? - Once you remember that, you can just step through the group axioms and show that they hold for (G,•) as well, provided they hold for (G,*).
For example: the associative property for (G,•) can be shown like this:
(a•b)•c = c*(b*a) = (c*b)*a = a•(b•c)
The first = holds, by definition of • in terms of *, the second = holds because (G,*) is a group, and the last = holds again by definition of • in terms of *.
Now you need to show the existence of a neutral element and of inverse elements.
Well, my guess is that the identity element of (G,•) is exactly the same as that of (G,*), so let's call the identity element of (G,*) simply e. Of course, that's just a hypothesis at this point, a lucky guess I hope. Now try to prove that this element e of G has the properties that are required if it is to qualify as an identity element of (G,•)!