Is the system consisting of the set 1 1 with the operation d

Is the system consisting of the set {-1, 1} with the operation division a group? If not, determine which properties are not satisfied. (Note that if the system is group, then all properties are satisfied.) Is the system a group? Yes No For each property, determine if it is satisfied. Closure Yes No Associativity Yes No Identity Yes No Inverse No

Solution

Let S denote the set {-1,1}. If S is to be a group, it must satisfy the 4 group axioms, namely closure, associativity, identity and invertibility.

We will check whether S is a group under multiplication.

Since (-1)*1 = 1*(-1) = 1, (-1)*(-1) = 1 and 1*1 = 1, the closure axiom is satisfied.

2. Associativity:

Since 1 [1*(-1)]= -1=[(1*1)]*(-1),(-1)*[ 1*(-1)]=1=[(-1)*1]*(-1), (-1)*[(-1)*(-1)]=-1=[ (-1)*(-1)]* (-1) and (1*1)*1 = 1 = 1*(1*1), hence the axiom of associativity is satisfied.

3. Identity:

1 is the (multiplicative) identity as 1*1 = 1 and 1*(-1) = (-1)*1 = -1.

4. 1 is the (multiplicative) inverse of itself and -1 is the (multiplicative) inverse of itself as 1*1 = 1 and, (-1)*(-1)= 1.

Thus, (S , *) satisfies all the group axioms and hence (S,*) is a group.