Quaternion Group: Determine that $i^4 = 1$.

Question

Suppose we are given the following presentation of the quaternion group:

$Q_8 = \langle i, j, k \ | \ i^2 = j^2 = k^2 = ijk\rangle$

Is it obvious that $i^4 = 1$?

Answer 1

From $k^2=ijk$ we get $k=ij$, and from $i^2=ijk$ we get $i=jk=jij$. Hence $i=jij=jjijj=j^2ij^2=i^2ii^2=i^5$, so $1=i^4$.