Let G be a group a Show that if h G satisfies hg g for some

Let G be a group. (a) Show that if h G satisfies hg = g for some g G, then h is the identity. b) Fix k G. Show that if hk = e, then h = k^-1.

Solution

Let G be a group. Show that if h G satisfies hg = g for some g G, then h is the identity.

(a). Let e be the identity element of G. Then, for all g , we have e g = g. Hence e g = hg and therefore, (e g) g^-1 = (hg )g^-1 or, e(g g^-1) = h(g g^-1) or, h e = g e or, h = .e

(b) Let k be an element of G and let hk = e. Then ( hk) k^-1 = e k^-1 or, h (k k^-1) = k^-1 or, h e = k^-1 or, h = k^-1.

Note: In a group, the binary operation is associative.