Decide whether the system, {−1,0,1} under multiplication, is a group.

Question

If it is not a group, identify which of the following properties it fails to satisfy: closure, associativity, identity, inverse.

A. Group.

B. Not a group; inverse.

C. Not a group; identity, inverse.

D. Not a group; closure.

Answer 1

I think it’s B, because multiplication of the elements produces an element in the group, demonstrating closure. Associativity is preserved because (0×1)×-1=0x(1×-1), for example. 1 is the identity element for multiplication. However, the inverse identity does not apply to zero because there is no element in the group such that 0×a=1.