Problem 3. Subgroups of quotient groups. Let G be a group and let H<G be a normal subgroup. Let K be a subgroup of G that contains H. (1) Show that there is a well-defined injective homomorphism i: K/ H G /H given by i(kH) = kH. By abuse of notation, we regard K/H as being the subgroup Imi < G/H consisting of all cosets of the form KH with k EK. (2) Show that every subgroup of G/H is...
Part B
Problem 2 Letº: G + H be a homomorphism, and let a E G be an element of finite order. a) Show that the order *(a) of q(a) is finite and that it divides (al. b) Show that if Q is an isomorphism, then 9(a)| = |al. Hint: Use that o(ak) = Q(a)k.
2. Let p: G -G be a surjective group homomorphism (a) Show that if G is abelian then G' is abelian. (b) Show that if G' is cyclic then there is a surjective homomorphism from (Z, +, 0) to G'. (Hint: use the fact that Z is generated by 1 and G' has a generator). (c) Use Part (a) and (b) to show that every cyclic group is abelian.
3. Let y: K + Aut(H) be a homomorphism. Write (k) = Ok. Let G be a group. A function d: K + H is called a derivation if dikk') = d(k) (d(k')). Show that d: K + H is a derivation if and only if V: K + H y K given by v(k) = (d(k), k) is a homomorphism. 4. Suppose that a: G + K is a surjective homomorphism and that 0: K + G is a...
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)...
4. Let G and H be isomorphic groups. Assume that every element of G has a square root (that is, for all IEG, there exists y E G such that I = y). Prove that every element of H has a square root.
Thanks
3. Suppose that S:G - H is a homomorphism of groups and that S is a normal subgroup of H. Then show that {r EG (2) ES) is a normal subgroup of NB: First show that {x EG f(x) E S} is a subgroups of G
Q3 (Due Wednesday 11 September—Week 7) Let (G, *) and (N,) be groups. Suppose that g Ha, is a homomorphism from from G to Aut(N)—that is, suppose that a, o ah = agh for all g, h E G. Let N a G denote the set N X G, and define a binary operation • on N a G by (m, g) + (a, b) = (m + ag(m), g * h). (1) Prove that (N a G, is a...
Let φ : G → H be any group homomorphism. Prove that φ is 1-1 if and only if ker(φ) = {e}.
5. Product groups: (a) Let G and G' be groups. Explain how one turns G × G, into a group by defining multiplication and identifying inverses and the identity element. (b) If G is an abelian group of order 30 and GZm x Zn how may possibilities are there for the whole numbers m and n? (Assume m S n for clarity and list the possibilities)