How do you show that the cyclic group $C_4$ is a subgroup of the Quaternion group?
Obviously the cyclic group $C_4$ is a subgroup of $Q_8$ but I was wondering how do you show this?
2026-03-29 11:50:13.1774785013
How do you show that the cyclic group $C_4$ is a subgroup of the Quaternion group?
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The number $i$ has order four in $Q_8$, since $i=i, i^2=-1, i^3=-i,$ and $i^4=(-1)^2=1$, so $C_4\cong\langle i\rangle,$ the subgroup generated by $i$.
NB: The group $C_4$ is isomorphic to a subgroup of $Q_8$, not equal to a subgroup.