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

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Let G be a group order 6, and H be a subgroup of G such that 1 < |H| < 6. Which one of the following options is correct?
1. G is always cyclic, but H may not be cyclic.
2. G may not be cyclic, but H is always cyclic.
3. Both G and H are always cyclic.
4. Both G and H may not be cyclic.

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Correct Answer - Option 2 : G may not be cyclic, but H is always cyclic.

Concept

According to the Lagrange theorem order of subgroups must divide the order of the group.

Property of group says if a group has prime order then it is cyclic.

Explanation:

Since the order of G is 6. Therefore  its subgroup may have order 1,2,3,6

H is one of its subgroups with condition 1< |H| <6 so H may be of order 2 or 3  which is prime

Hence H must be cyclic

The order of G is 6 which is not prime and hence it may or may not  be cyclic

Therefore option 2 is correct