# Group Theory – Determine if (S,*) is a group

(Phils)

**Abstract Algebra - Group Theory**Determine if (S,*) is a group (verify the properties of a group are satisfied).

a. S= The set of real numbers; the binary operation * is defined by a*b= (a+b)/2, a,b are element of S

b. S= The set of integers Z; the binary operation * is the ordinary operation subtraction.

c. S= The set 3Z = {3n| n E Z};the binary operation * is the ordinary operation addition.

d. S= The set G={0,1,2}; the binary operation * is defined by a*b = |a-b|, a and b are element of S.

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I know that in order for the set S to be a group, it must satisfy the four condition( closure, associative, identity, and inverse)

for Qa

1.it satisfies condition 1 because for every a*b, there is always c that is an element of S but i can't prove conditions 2,3 and 4

For Qb.

1. yes it is closed under subtraction because for evry a*b, there is c that is belong to the set S.

2. Subtraction is not associative on Z

condition 3 and 4 confuse me because accrding to the defintion of identity a*e=a. Based on this definition, the identity is 0. But if the identity is zero then 0 must be the difference of any integer and its inverese. This does not happen (ex. 3-(-3) =6 not 0

Does this mean there is no identity and inverse?

I don't know how to start question c

Qd.

yes it is closed because for every A*b, there is c thetis an element of S

Idon't know how to show that it is associative

yes it has an identity. It's 0

the inverses are:

the inverse of 0 is 0

the inverse of 1 is 1

the inverse of 2 is 2