11. Let G be a nonabelian group of order 8. (a) Prove that G must have an element of order 4, but none of order 8. (b) Leta be an element of order 4, and let N = (a). Show that there exists an element 1) such that G = N U Nb. (c) Show that either [)2 = e or b2 = a2. (Since N is normal, consider the order of N b in G/ N.) ((1) Show that bub1 has order 4 and must be equal to 0.3. 4 (e) Conclude that either G \"=" D4 or else G is determined by the equations a = e, ba 2 a319, [)2 = a2. Review Example 3.3.7 (the quatemion group) to verify that the second case can occur