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Let G be a group of order 6, and H be a subgroup of G such that 1 < |H| < 6. Which one of the following options is correct?

1. Both G and H are always cyclic.
2. There exist groups G and H such that G is cyclic and H is not cyclic.
3. There exist groups G and H such that G is not cyclic and H is cyclic.
4. Both S1 and S2 are false.

Correct answer: There exist groups G and H such that G is not cyclic and H is cyclic.

Solution

A group of order 6 can either be cyclic or isomorphic to the symmetric group S3, which is not cyclic. If G is isomorphic to S3, it has a subgroup of order 3 that is cyclic, demonstrating that H can be cyclic while G is not.

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