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22. 23. 24. 25. 26. 27. 28. 29. 31}. 31. appeared as 1, 9, 16, 22, 53, 74, T9, 81. Which integer was left out? (This really happened!) Let G be a group with the property that for any x, y, z in the group, xy : ax implies )7 : z. Prove that G is Abelian. (\"Left-right cancel- lation\" implies commutativity.) (Law of Exponents for Abelian Groups) Leta and b be elements of an Abelian group and let a be any integer. Show that (at-0\" : 51%". Is this also true for non-Abelian groups? (Socks-\"Shoes Property) Draw an analogy between the statement (am1 : 1-51071 and the act of putting on and taking off your socks and shoes. Find distinct nonidentity elements a and b from a non-Abelian group such that (am1 : a'1 191. Find an example that shows that in a group, it is possible to have (am2 at 192 erg. What would be an appropriate name for the group property (abcfl = c1 b'1 (11? Prove that a group G is Abelian if and only if (am1 I ISI1bl for all a and b in G. Prove that in a group, (.sfl)'l : a for all a. For any elements a and b from a group and any integer n, prove that (trlbw" : a'lb"a. If :11, a2, . . . , an belong to a group, what is the inverse of at\": - - - an? The integers 5 and 15 are among a collection of 12 integers that form a group under multiplication modulo 56. List all 12. Give an example of a group with 105 elements. Give two examples of groups with 44 elements. Prove that every group table is a Latin squarel; that is, each ele