   Chapter 3.1, Problem 13E

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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 with operation addition.

To determine

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

Explanation

Given information:

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

Explanation:

Let us check if the set of all multiples of a positive integer n with operation addition 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=(p+q)n.

As p,q, p+q.

As x+y is also a multiple of n, x+yS

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