If G and H are groups with identities 16 and ¹H, respectively, and f:G ---> H is a homomorphism, then the kernel öf f is a. f(¹G) b. f(G) c. f-¹ (H) d. f-¹(1g) e. ¹H
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- Prove or disprove that H={ hGh1=h } is a subgroup of the group G if G is abelian.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.Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .
- 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.Suppose that G and G are abelian groups such that G=H1H2 and G=H1H2. If H1 is isomorphic to H1 and H2 is isomorphic to H2, prove that G is isomorphic to G.15. Prove that if for all in the group , then is abelian.