9. Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no subgroup of order 30. OC
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- Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.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 12.43. Suppose that is a nonempty subset of a group . Prove that is a subgroup of if and only if for all and .
- 15. Assume that can be written as the direct sum , where is a cyclic group of order . Prove that has elements of order but no elements of order greater than Find the number of distinct elements of that have order .15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units . Sec. 16. For an integer , let , the group of units in – that is, the set of all in that have multiplicative inverses, Prove that is a group with respect to multiplication.