By N. Jacobson
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12. 1, if ISI = 2, then composition, as an operation on M(S), is not commutative. However, Sym(S) is Abelian. Explain. 13. Prove that if a is a k-cycle with k > 2, then is o a is a cycle if k is odd. 14. 7 (after first writing a in one-row form). Check your answer using only two-row forms. 15. If G is a group with operation *, and a, b E G, then b * a * b-' is called a conjugate of a in G. 7 to help compute the number of conjugates of each 3-cycle in S. (n > 3). 16. Assume that a and 0 are disjoint cycles representing elements of S,,, say a = (aI a2 .
21. Give an example of sets S, T, and U and mappings a : S -> T and is onto, but P is not onto. 22. Give an example of sets S, T, and U and mappings a : S -+ T and is one-to-one, but a is not one-to-one. 23. 24. Prove that if : S -* T, y : S -> T, a : T - U, a is one-to-one, and a o = y. 25. 23. 26. 24. 27. Assume that a : S T and fi : T -> U. 2 to prove each of the following statements. (a) If a and 0 are invertible, then fl o a is invertible. (b) If fi o a is invertible, then $ is onto and a is one-to-one.
This example illustrates why, in general, we should specify the operation, not just the set, when talking about a group. 6. If S is any nonempty set, then the set of all invertible mappings in M(S) is a group with composition as the operation. 1(b). We shall return to groups of this type in the next section. 7. Let p denote a fixed point in a plane, and let G denote the set of all rotations of the plane about the point p. 1 we observed that composition is an operation on this set G, and we also verified everything needed to show that this gives a group.
Basic Algebra II by N. Jacobson