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Groups and FieldsChallengeCosets and Lagrange's Theorem

Cosets and Lagrange's Theorem · Challenge problem

Q8={±1,±i,±j,±k}Q_8=\{\pm1,\pm i,\pm j,\pm k\} is the quaternion group, order 88. Lagrange allows subgroups of order 1,2,4,81,2,4,8. How many subgroups does Q8Q_8 actually have, counting all orders?