. Let N be a finite group and let H be a subgroup of N. If [H| is odd and [N:H] = 2, prove that the product of all.of the elements of N, in any order, cannot belong to H.
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![4. Let N be a finite group and let H be a subgroup of N. If |H| is odd and [N:H] = 2, prove that
the product of all of the elements of N, in any order, cannot belong to H.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F174d7d89-86f6-4fd3-817b-704de3f40cca%2F109e05ab-0521-4fef-b70f-187af95a109e%2Fk0g930k_processed.png&w=3840&q=75)
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- Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?43. Suppose that is a nonempty subset of a group . Prove that is a subgroup of if and only if for all and .Exercises 30. For an arbitrary positive integer, prove that any two cyclic groups of order are isomorphic.
- 27. Suppose is a normal subgroup of order of a group . Prove that is contained in , the center of .23. Prove that if and are normal subgroups of such that , then for all15. 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.(See Exercise 31.) Suppose G is a group that is transitive on 1,2,...,n, and let ki be the subgroup that leaves each of the elements 1,2,...,i fixed: Ki=gGg(k)=kfork=1,2,...,i For i=1,2,...,n. Prove that G=Sn if and only if HiHj for all pairs i,j such that ij and in1. A subgroup H of the group Sn is called transitive on B=1,2,....,n if for each pair i,j of elements of B there exists an element hH such that h(i)=j. Suppose G is a group that is transitive on 1,2,....,n, and let Hi be the subgroup of G that leaves i fixed: Hi=gGg(i)=i For i=1,2,...,n. Prove that G=nHi.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.