Suppose that G,H are groups with identities eG,eH,
respectively.
Define f: G x H→G by setting, for all g in G and h in H,
f(g,h) = g.
a) Prove that f is a homomorphism.
b) Prove that f is surjective.
c) Prove that ker(f) = {(eG,h) | h in H}.
We need at least 10 more requests to produce the answer.
0 / 10 have requested this problem solution
The more requests, the faster the answer.
Suppose that G,H are groups with identities eG,eH, respectively. Define f: G x H→G by setting,...
Let a : G + H be a homomorphism. Which of the following statements must necessarily be true? Check ALL answers that are necessarily true. There may be more than one correct answer. A. If kera is trivial (i.e., ker a = {eg}), then a is injective. B. If the image of a equals H, then a is injective. C. The first isomorphism theorem gives an isomorphism between the image of a and a certain quotient group. D. The first...
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...
I. Functions and Isomorphisms. Let G be a group and let a EG be any non-identity element (so a #e). Define a function f : GG so that, for any r EG, f(x) = (xa)-1 (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer. (c) Is f an isomorphism? Prove your answer.
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...
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag to be the collection of all equivalence classes. Prove that the function g : A ! E deÖned by g (x) = Ex is...
4. Suppose that G = (V, E) and H = (W, F) are two graphs. Define a new graph KG,h by: The vertex set of KG,H is VUW. xy is an edge of KG,H if and only if either: x,y € E, or x, y € F, or x € V and ye W. Prove that x(KG,H) = x(G) + x(H).
7. [5 marks] Suppose that f(x), g(x), and h(x) are functions such that f(x) is O(g(x)) and g(a) is O(h(x)). Prove that f(x) is O(h(x)) 7. [5 marks] Suppose that f(x), g(x), and h(x) are functions such that f(x) is O(g(x)) and g(a) is O(h(x)). Prove that f(x) is O(h(x))
Please solve all parts in this problem neatly 3. Let f(x, y, ). g(y,z) and h(x,y,z) be C2 scalar functions. Prove the following identity: (a) By direct calculation (without using the vector identities) ( b) Using the vector identities. Clearly state which identities you have used . 3. Let f(x, y, ). g(y,z) and h(x,y,z) be C2 scalar functions. Prove the following identity: (a) By direct calculation (without using the vector identities) ( b) Using the vector identities. Clearly state...
III. Properties of Isomorphisms. Let G and H be isomorphic groups and suppose that 0 : G + H is an isomorphism. Assume L is a subgroup of H and define K = {g € Gº(9) EL}. Prove that K is a subgroup of G.