- Show that the following subset is a subgroup. H = {o e S, l0(n) = n} S,
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- (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.42. For an arbitrary set , the power set was defined in Section by , and addition in was defined by Prove that is a group with respect to this operation of addition. If has distinct elements, state the order of .Find subgroups H and K of the group S(A) in example 3 of section 3.1 such that HK is not a subgroup of S(A). From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of all permutations defined on A.
- 18. If is a subgroup of , and is a normal subgroup of , prove that .True or False Label each of the following statements as either true or false. 8. The set of all odd permutations in is a subgroup of .Let n be appositive integer, n1. Prove by induction that the set of transpositions (1,2),(1,3),...,(1,n) generates the entire group Sn.