Prove that G={−1,1,i,−i} is a group under multiplication.

First need to show that G is indeed closed under the operation *

we have 1∗1=1 where 1∈G

we have −1∗−1=1 where 1∈G

we have 1∗−1=−1 where −1∈G and −1∗1=−1∈G

we have 1∗i=i and i∗1=i where i∈G

we have −1∗i=−i and i∗−1=−i where −i∈G

let k∈N then i2k=−1 where −1∈G

Finally let k∈N then we have i2k+1=−i where −i∈G

So, all possible outcomes from every combination of multiplication between any elements yields an element in G.