Either give an example (with explanation), or explain why the following is not possible. (a) A group with no normal subgroups. (b) An element in Z15 × Z42 × Z10 of order 450. (c) A homomorphism from D8 into D8 whose kernel is {1, r, r2, r3}.

Either give an example (with explanation), or explain why the following is not possible. (a) A group with no normal subgroups. (b) An element in Z15 × Z42 × Z10 of order 450. (c) A homomorphism from D8 into D8 whose kernel is {1, r, r2, r3}.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.8: Some Results On Finite Abelian Groups (optional)

Problem 15E: 15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that...

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Question

Either give an example (with explanation), or explain why the following is not
possible.

(a) A group with no normal subgroups.

(b) An element in Z15 × Z42 × Z10 of order 450.

(d) An infinite group that is finitely generated.

(e) A permutation in S11 that is both even and odd.

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