(a). H, K≤ G and HGH* K≤G. (b). HG and KAG⇒ H✩ KAG.
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- Find the normalizer of the subgroup (1),(1,3)(2,4) of the octic group D4.24. Let be a group and its center. Prove or disprove that if is in, then and are in.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?
- (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.Prove that Ca=Ca1, where Ca is the centralizer of a in the group G.23. Prove that if and are normal subgroups of such that , then for all
- Let A={ a,b,c }. Prove or disprove that P(A) is a group with respect to the operation of union. (Sec. 1.1,7c)45. Let . Prove or disprove that is a group with respect to the operation of intersection. (Sec. )Show that * defined on Z by a*b=|a+b| is not a group. (Hint: identify and show the group property that is not satisfied).