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16E

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Abstract Algebra: An Introduction

Found in: Page 246

Book edition 3rd

Author(s) Thomas W Hungerford, David Leep

Pages 608 pages

ISBN 9781111569624

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Short Answer

Suppose G is a cyclic group $< a >$ and $| a |$ =15 . If role="math" localid="1651649969961" $K = < a^{3} >$ , list all the distinct cosets of K in G .

The distinct cosets of K in G are K e, K a, K a² .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Group G

Given that G is a cyclic group $<a>$ of order 15 which are {1, a ,a²,a³,a⁴,a⁵,a⁶,a⁷a,⁸,a⁹,a¹⁰,a¹¹,a¹²,a¹³,a¹⁴} .

Distinct cosets of K in G

Let K = $<a^{3}>$ be the subgroup of G then, the cosets of K can be separated as Ke{1, a³, ⁵,a⁶,a⁹,a¹²}, Ka² = {a²,a⁵,a,⁸,a¹¹ ,a¹⁴} . } and .

Therefore, the distinct cosets of K in G are Ke, Ka, Ka² .

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