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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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In Exercises 1 14 , decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition 3.1 that fails to hold.

The set of all multiples of a positive integer n is group with operation multiplication.

To determine

Whether the set of all multiples of a positive integer n is a group with operation multiplication.

Explanation

Given information:

The set of all multiples of a positive integer n with operation multiplication.

Explanation:

Let us check that the set of all multiples of a positive integer n with operation multiplication is a group or not by using the definition given below:

Suppose the binary operation is defined for element of a set G. The set G is a group with respect to , provided the following conditions hold.

1. G is closed under . That is xG and yG imply that xy is in G.

2. is associative. For all x,y,z in G, x(yz)=(xy)z.

3. G has an identity element e. There is an e in G such that xe=ex=x for all xG.

4. G contains inverses. For each aG, there exists bG such that ab=ba=e.

Let n be any positive integer.

The set of all multiples of n is S={...,3n,2n,1n,0,1n,2n,3n,...}

First condition:

Let x,yS such that x=pn,y=qn where p,q

x×y=pn×qn=(pqn)n

As p,q,andn+, pqn

x×y is also a multiple of n, x×yS

S is closed under multiplication

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