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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem
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In Exercises 15 and 16 , the given table defines an operation of multiplication on the set S = { e , a , b , c } . In each case, find a condition in Definition 3.1 that fails to hold, and thereby show that S is not a group.

See Figure 3.6 .

× e a b c
e e a b c
a a b a b
b b c b c
c c e c e

To determine

Which condition fails to show that S is not a group with operation multiplication.

Explanation

Given information:

The set S={e,a,b,c} with operation multiplication and table:

×eabceeabcaababbbcbcccece

Explanation:

Let us check if the given table 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

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