Let G be a group with GG Let H

Let G be a group with |G|=G

Let H<G with HG then H is cyclic .True or False?

Solution

True

If H is a subgroup of a cyclic group, then H is necessarily normal since every subgroup of an abelian group is normal*. As such, we know there exists a group G1 and a surjective group homomorphism :GG1 such that ker()=H From the isomorphism theorem, we know that G1G/H

So now the problem boils down to whether the homomorphic image of a cyclic group is cyclic. That is rather immediate: if gg generates G, then (g) must generate G1. To see this, consider any aG1. Then a=(g^n) for some n, and further (g^n)=(g)^n.