Solve

G such that xax = b if and only if ab = c2 for some element c in G. 11. Prove that if a is the only element of order 2 in a group, then a lies in the center of the group. 12. Let G be the plane symmetry group of the infinite strip of equally spaced H's shown below. H H H H H Axis 1 Axis 2 Let x be the reflection about Axis 1 and let y be the reflection about Axis 2. Calculate Ixl, lyl, and lxyl. Must the product of elements of finite order have finite order? (This exercise is referred to in Chapter 27.) 13. What are the orders of the elements of D,? How many elements have each of these orders? 14. Prove that a group of order 4 is Abelian. 15. Prove that a group of order 5 must be cyclic. Prove that an Abelian group of order 6 must he evelie