   Chapter 3.1, Problem 24E

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# In Exercises 17 − 24 , let the binary operation ∗ be defined on ℤ by the given rule. Determine in each case whether ℤ a group with respect to is ∗ and whether it is an abelian group. State which, if any, conditions fail to hold. x ∗ y = 2 x − y

To determine

Whether is a group with respect to where xy=2xy and whether is an abelian group. State which, if any, conditions fail to hold.

Explanation

Given information:

The binary operation on such that xy=2xy.

Explanation:

Suppose the binary operation is defined for the 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.

Condition for an abelian group:

G is called commutative group or an abelian group, if is commutative. That is, xy=yx for all x,y in G.

First, check if is a group under the binary operation.

First condition:

Let x,y

2xy

xy

is closed under the binary operation.

Second condition:

Let x,y,z.

x(yz)=x(2yz)=2x(2yz)=2x2y+z

(xy)z=(2xy)z=2(2xy)z=4x2yz

As x(yz)(xy)z, is not associative with respect to the binary operation

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