**Unformatted text preview: **04/26/2002 FRI 12:27 FAX 6434330 MOFFITT LIBRARY .001 George M. Bergman Fall 2001, Math 113, Sec. 3 28 Sept, 2001
70 Evans Hall First Midterm 11:10—12:00 1. (24 points, 8 points apiece) Find the following. (a) The remainder r when 2953 — 103 is written in the form 19- 9 + r with O s r S 9.
(b) (1, 2,3,4) (2,4, 5T1, expressed as a product of disjoint cycles in S6.
(0) The order of the cyclic subgroup ((1,2)(3,4)(5,6,7)) of S10. 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.) _
(a) An integer x such that x a 27 (mod 100) and x E 59 (mod 101). (You need not write it
out explicitly; you may instead giVe an arithmetic expression whose value must have that property.) 03) An element of order 10 in 89.. (c) An element of order 2 in a ﬁnite group of odd order. (d) An isomorphism 1;! between Z3 and a subgroup of S3. 3. (20 points) Deﬁne what is meant by a group (G, -).
(You need not use the exact wording of the deﬁnition in the text as modiﬁed in my
corrections; but for full credit your answer must clearly express all the conditions in that deﬁnition, and should avoid the type of poor wording that my corrections changed.) 4. (20 points) Let G be a group. Show that for all a, [3, CE G the equation axb = c has a
unique solution in G. (This means showing both that an element x exists which satisﬁes the equation, and that there is only one such element.) ...

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