2. If H and K are subgroups of an abelian group G, then HK = {hk | h e H and k e K} is a subgroup of G.
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- Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?If a is an element of order m in a group G and ak=e, prove that m divides k.Let H1 and H2 be cyclic subgroups of the abelian group G, where H1H2=0. Prove that H1H2 is cyclic if and only if H1 and H2 are relatively prime.
- Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic group of order n. Prove that G has elements of order 12 but no element of order greater than 12. Find the number of distinct elements of G that have order 12.Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.