5. (5) If f:G H is a homomorphism, H is abelian and N is a subgroup...
(a) Show that if and are subgroups of an abelian group , then is a subgroup of . (b) Show that if and are normal subgroups of a group G then is a normal subgroup of (4)(20 points) (a) Show that if H and K are subgroups of an abelian group G, then HK = {hk | h € H, k € K} is a subgroup of G. (b) Show that if H and Kare normal subgroups of a group G, then HNK is...
4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G 4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G
#7 7 Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian. If G is cyclic, prove that G/H must also be cyclic. 8.
9) A group G is called solvable if there is a sequence of subgroups such that each quotient Gi/Gi-1 is abelian. Here Gi-1 Gi means Gi-1 is a normal subgroup of Gi. For example, any abelian group is solvable: If G s abelian, take Go f1), Gi- G. Then G1/Go G is abelian and hence G is solvable (a) Show that S3 is solvable Suggestion: Let Go- [l),Gı-(123)), and G2 -G. Here (123)) is the subgroup generated by the 3-cycle...
Abstract algebra A. Assume G is an abelian group. Let n > 0 be an integer. Prove that f(x) = ?" is a homomorphism from Got G. B. Assume G is an abelian group. Prove that f(x) = 2-1 is a homomorphism from Got G. C. For the (non-abelian) group S3, is f(x) = --! a homomorphism? Why?
1- (2,5+2,5 mark) Consider in GL(2, Q), the subset (a=1 or a=-1),bez Prove that H, with multiplication, is a subgroup of GL(2,Q) a) Is the function b) an homomorphism of groups? Justify your answer 2 (3 marks) Let G be a group and a E G an element of order 12. Find the orders of each of the elements of (a) 3- (1+1,5 marks) Let G be a group such that any non-identity element has order 2. Prove that a)...
1. Let G - Z. Let H - {0,3,5,9) be a subgroup of (you do not need to prove this is a subgroup of G). Prove that G/l is a valid quotient group. Explain what the elements of G/H are and what the group operation is. 2. Let G be a group and H a normal subgroup in G. I E H for all IEG, then prove that G/H is abelian
I help help with 34-40 33. I H is a subgroup of G and g G, prove that gHg-1 is a subgroup of G. Also, prove that the intersection of gH for all g is a normal subgroup of G. 34. Prove that 123)(min-1n-)1) 35. Prove that (12) and (123 m) generate S 36. Prove Cayley's theorem, which is the followving: Any finite group is isomorphic to a subgroup of some S 37. Let Dn be the dihedral group of...
Prove that if H and N are subgroups of G with Na normal subgroup of Gthen N nHis normal in H but not neccesarly in G Prove that if H and N are subgroups of G with Na normal subgroup of Gthen N nHis normal in H but not neccesarly in G
7. Let A be a Abelian group of finite order n and let m be a natural number. Define a map Om : A + A by Om(a) = a". Prove that Om is a homomorphism of A and identify the kernel of øm. Determine when om is an isomorphism.