   Chapter 3.1, Problem 27E

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# In Exercises 25 − 32 , decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state all of the conditions in Definition 3.1 that fail to hold. If it is a group, state its order.The set { [ 1 ] , [ 2 ] , [ 3 ] } ⊆ ℤ 4 with the operation multiplication.

To determine

Whether the set {,,}4 is a group with operation multiplication. State which, if any, conditions fail to hold. If it is a group, then state its order.

Explanation

Given information:

The set {,,}4 with operation multiplication.

Explanation:

Definition of a group:

Suppose the binary operation is defined for element of set G. Then 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.

Definition of order of group:

The number of elements in group G is called the order of G, and it is denoted by either o(G) or |G|.

To check the conditions of a group for the set E={,,}4 with operation multiplication where 4={,,,}, use the table form as below:



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