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Math 113 - Fall 2002 - Bergman - Midterm 2

A First Course in Abstract Algebra, 7th Edition

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Unformatted text preview: 05/12/2003 MON 16:23 FAX 6434330 MOFFITT LIBRARY .001 George M. Bergman Fall 2002, Math 113, Sec. 5 28 Oct, 2002 5 Evans Hall Second Midterm 3:10—4:00 1. (27 points, 9 points each.) Find the following. If the answer to a question is a set, you should give it by listing or describing its elements in set brackets, { }. (a) The kernel of the homomorphism from Z to 010 (the group of symmetries of a pentagon) taking each :16 Z to rotation by n(4 113/5) radians. (b) The coset of A3 in S3 that contains (12). (c) The number of fixed points of 03, if 0' is an element of Sn whose complete cycle decomposition consists of a cycles of length 3, 1) cycles of length 2, and n — 3a — 219 cycles of length 1. 2. (36 points; 9 points each.) For each of the items listed below, either give an example, or give a brief reason why no example exists. (If you give an example, you do not have to prove that it has the property stated. Examples should be specific for full credit; i.e., even if there are many objects of a given sort, you should name one.) (a) A simple non—cyclic group. (b) A subgroup of Z X Z that is not normal. (0) An injective (i.e.-, one-to-one) homomorphism f : Z ~—> Rx. (Recall that [Rx denotes the group of nonzero real numbers under multiplication.) (d) A group G and a subgroup H, such that H is not the kernel of any homomorphism with domain G. 3. (14 points.) Let G and H be groups, f: G——> H an injective (i.e., one—to-one) homomorphism, and 3E G an element of finite order n. Show that f(g) also has order n. ' 4. (14 points.) Let G be a group. Recall that Z(G), the center of G, means {zEG: V gE G, .23 = gz}. Show that Z(G) is a subgroup of G. (Rotman describes this as “easy to see”. I am asking you to supply the details.) 5. (9 points.) Let G be a group which acts on a set X, and let x,ye X. Show that if O(x) and O(y) have an element in common, then they are equal. (Recall that O(x) denotes {gx : g6 G}. The result you are to prove is part of a result proved by Rotman, that X is the disjoint union of the orbits. Hence you may not call on that result in proving this.) ...
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