4023f11ps4

4023f11ps4 - H = a 4 4 Suppose K is a proper subgroup of H...

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Homework #4 Due: September 21, 2011 1. Two results concerning groups are (1): If H and K are subgroups of G , then H \ K is a subgroup (Theorem 0.4.4), and (2): Every subgroup of a cyclic group is cyclic (Theorem 0.4.5). For each of the following subgroups H and K of a cyclic group G = [ a ], identify H \ K by giving a generator. That is, write H \ K = [ b ] for some b 2 G . (a) G = Z , H = 6 Z , K = 45 Z . (b) G = [ a ] with o ( a ) = 20, H = [ a 14 ], K = [ a 15 ]. 2. Find all of the subgroups of each of the following groups. (a) Z 18 (b) G = [ a ] where o ( a ) = 28. (c) Z / 13 . Hint: Show flrst that Z / 13 is cyclic with generator 2. 3. (a) Find all of the left cosets of the subgroup H = 4 Z 8 of the group G = Z 8 . Recall that for additively written groups, such as Z 8 , the cosets are written additively in the form a + H = f a + h : h 2 H g . (b) Suppose that G = [ a ] with o ( a ) = 12. Find all of the cosets of the subgroup
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Unformatted text preview: H = [ a 4 ]. 4. Suppose K is a proper subgroup of H and H is a proper subgroup of G . ( H is a proper subgroup of G means that H is a subgroup but H 6 = G .) If j K j = 42 and j G j = 420, what are the possible orders of H . 5. Find the value of the Euler phi-function ` ( n ) for (a) n = 97; (b) n = 8800. 6. Use Euler’s Theorem (theorem 0.6.1) to flnd a number a with 0 • a < 73 with a · 9 794 (mod 73) : 7. For each part, flnd the smallest positive x that solves the given simultaneous congru-ences. (a) x · 5 (mod 7) and x · 5 (mod 9) (b) x · 5 (mod 5) and x · 2 (mod 12) and x · 8 (mod 13) Math 4023 1...
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This document was uploaded on 12/28/2011.

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