7. Suppose G is a finite group and let n be a factor of |G|. Suppose N≤ G is the only subgroup of G of size n. Prove that N is a normal subgroup of G.
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A: Given,C1 : x2 +y2 =16 , C2 : 4-(x+2)2, and C3 : -4x-4
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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?Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.34. Suppose that and are subgroups of the group . Prove that is a subgroup of .
- 15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.43. Suppose that is a nonempty subset of a group . Prove that is a subgroup of if and only if for all and .10. Suppose that and are subgroups of the abelian group such that . If is a subgroup of such that , prove that .
- Let be a subgroup of a group with . Prove that if and only if(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.Find all Sylow 3-subgroups of the symmetric group S4.
- 22. If and are both normal subgroups of , prove that is a normal subgroup of .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 .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.)