GATE 2020 MA – Question 25
Find the number of cyclic subgroups of the quaternion group $Q_8=\langle a,b\mid a^4=1,a^2=b^2,ba=a^3b\rangle$.
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Correct answer: 5 to 5
Explanation
The trivial subgroup and the order-two subgroup {1,−1} are cyclic. The six elements of order four generate three distinct subgroups, {±1,±i}, {±1,±j} and {±1,±k}. There are no other cyclic subgroups because Q₈ has no element of order eight. The total is 1 + 1 + 3 = 5.