Let ø be a homomorphism from a group G to a group H. Let K be a subgroup of H. Prove that ø (K) = { g e G : ø(g) e K } is a subgroup of G.
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- Let G be a group of order pq, where p and q are primes. Prove that any nontrivial subgroup of G is cyclic.Let H be a normal cyclic subgroup of a finite group G. Prove that every subgroup K of H is normal in G.27. Suppose is a normal subgroup of order of a group . Prove that is contained in , the center of .
- 34. Suppose that and are subgroups of the group . Prove that is a subgroup of .Let H be a torsion subgroup of an abelian group G. That is, H is the set of all elements of finite order in G. Prove that H is normal in G.Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.
- Exercises 31. Let be a group with its center: . Prove that if is the only element of order in , then .24. The center of a group is defined as Prove that is a normal subgroup of .Let H be a subgroup of the group G. Prove that if two right cosets Ha and Hb are not disjoint, then Ha=Hb. That is, the distinct right cosets of H in G form a partition of G.