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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?15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.If a is an element of order m in a group G and ak=e, prove that m divides k.
- 9. Find all elements in each of the following groups such that . under addition. under multiplication.14. Find groups and such that and the following conditions are satisfied: a. is a normal subgroup of . b. is a normal subgroup of . c. is not a normal subgroup of . (Thus the statement “A normal subgroup of a normal subgroup is a normal subgroup” is false.)5. For any subgroup of the group , let denote the product as defined in Definition 4.10. Prove that corollary 4.19:
- True or False Label each of the following statements as either true or false. 4. If a subgroup of a group is cyclic, then must be cyclic.18. If is a subgroup of , and is a normal subgroup of , prove that .Find groups H and K such that the following conditions are satisfied: H is a normal subgroup of K. K is a normal subgroup of the octic group. H is not a normal subgroup of the octic group.