Let Q be the set of eight elements plusminus 1 plusminus i p

Let Q be the set of eight elements, {plusminus 1, plusminus i, plusminus j, plusminus k}. We define the binary operations by the rules ij = -ji = k, jk = -kj = i, ki = -ik = j, i^2 = j^2 = k^2 = -1, -1i = -I, and (-1)^2 = 1. The rules with i, j, and k can illustrated by using the diagram below. The center Z(G) of a group G is the subset of elements in G that commute with every clement of G: Z(G) = {a epsilon G|ax = xa for x epsilon G} Find the centre of the quaternion group, Z(Q). Show that Z(Q) is a normal subgroup of Q. Construct the Caley table for Q/Z(Q). Is Q/Z(Q) a group? Is it abelian?

Solution

We have given that
The centre of a group is the set of all elements of the group that commute with every element in the group:
Z(G) = { a in G | ab = ba for all a in G}.

In the case of the quaternions Q = {1,-1,i,-i,j,-j,k,-k},
with i^2 = j^2 = k^2 = ijk = -1,
since i, j, and k (or their opposites) do not commute with each other,

Since we have given that ij = -ji, jk = -kj , ki = -ik
we conclude that center is Z(Q) = {-1,1},
since these two elements clearly do commute with every elment in Q.