2. Let G and H be any two groups. Define the set G x {e} = {(g, eμ) | gЄG} which is a subset of G × H. a) Show that the inverse of (a, b) E G x H is (a¹, b¹). b) Prove that G x {e} is a normal subgroup of G × H. c) Show that f: G x H→ H defined by f((g, h)) = h is a surjective homomorphism. d) Find the kernel of f. e) Apply the First Isomorphism Theorem (of groups) to the function f.

Q: SUCH THAT LET H BE A PROPER SUBGROUP OF G V x,y € G-H, xy EH. PROVE THAT HAG.

A:

Q: Chapter 2, exercise 11.6: Let G be a group that contains normal subgroups of orders 3 and 5,…

A:

Q: subgroups of G. Let x,y e G. We define the relation x ~ y by the following definition: x ~ y iff y =…

A:

Q: (6) Define the center of a group G as the subset Z = {z EG: zg = gz for all g € G}. Prove that Z is…

A:

Q: (a, b | a group of degree 3. Let G 10.1.2. Let Dg b? = e, ba a3b), and let S3 be the symmetric (b) ×…

A:

Q: 15. Let G be a group and let g be in G. Define a function :G → G by ,(x) = gxg Show that :G→G is…

A: Let G be a group and let g be in G . Define a function ϕg : G→G by ϕgx=gxg-1. To prove: ϕg : G→G is…

Q: ts] Let G be a group, and let H be a subgroup of G. Show that if G is cyclic then H is also cyclic.…

A:

Q: Chapter 2, exercise 11.1: Let x be an element of order r of a group G and let y be an element of a…

A:

Q: 5. Let a E G be fixed. Define y: G → G by p(x) = axa-1 for all x E G. Prove that p is a…

A: To find - Let a∈G be fixed. Define ψ:G→G by ψx=axa-1 for all x∈G. Prove that ψ is a homomorphism.

Q: Let a € Aut G and let H = {g € G: ga = g}. Prove that H is a subgroup of G, it is called the fixed…

A: Since provided that  α∈Aut(G)  and            H=g∈G : gα=g We have to prove that  H is a subgroup of…

Q: Let G be a group and let H and K be normal subgroups such that HnK = {e}. Let : G→G/HxG/K be the map…

A: Let  be  a  group  and let and  be  normal subgroups   such  that .Let   be  the map We have   to…

Q: 4. Recall that Z(G) = {r € G| gr = rg, Vg E G}. Show that Z(G) is a normal subgroup of G.

A:

Q: Let G be a group and N is a normal subgroup of G. Let f be a mapping from G to H/N defined by f(x) =…

A: Given : Let G be a group and N is a normal subgroup of G and Let f be a mapping from G to H/N…

Q: Chapter 2, exercise 10.2(a): Let H and K be subgroups of a group G. Prove that the intersection…

A:

Q: [:] a b 1. Let T be a group of all invertible 2 ×2 matrices of the form C 1 x Where a,b,c €R and ac…

A:

Q: Let G = GL2(R). {[ {[ а (a) Prove that T = | a, c, d e R, ad is a subgroup of G. d a (b) Prove that…

A:

Q: 1. Let A be a non-empty set and (G, .) be a group. Let F be the set of all functions from A to G.…

A: Let A be a non empty set. For f,g∈F with the operation f*gx=fx.gx To prove that F,* is a group.…

Q: Let n > 2 be an integer, and let X C Sn × S, be the set X = {(ơ, T) | 0(1) = T(1)}. Show that X is…

A:

Q: Let G be a group and N is a normal subgroup of G. Let f be a mapping from G to H/N defined br f(x) =…

A:

Q: Theorem. Let F be a field. The set A(F) of affine transformations on F, given by x → ax + b, where…

A:

Q: Let G be a group and H a normal subgroup of G. Show that if x.V EG Such that xvEH then X,y xyƐH yx…

A: The solution is given as

Q: Let G be a group and let H and K be normal subgroups such that HnK={e}. Let : G→G/HxG/K be the map…

A: The objective of this question is to prove that the kernel of the map φ: G → G/H × G/K, where φ(g) =…

Q: SIZ): Set of all fiZ-→Z which are both one-one S(Z): on ti and S(Z) is a group under composition of…

A: Given, S(ℤ)=f|f:ℤ→ℤ, 1-1, onto is a group under composition of function. And H=fn:ℤ→ℤ|fn(x)=n+x.…

Q: Let G=U(15) and H={1,4,7,13} be a subgroup of G. The distinct left cosets of H in G are: * {Н, 7H}…

A:

Q: Let G be a group and let H and K be normal subgroups such that Hnk= G/H x G/K be the map o(g) = (Hg,…

A: A mapping from a group (G,o) to a group (G',o') is called a homomorphismif f(aob)=f(a)o'f(b)where a,…

Q: Let G and H be groups. Let p : G → H be a homomorphism and let E < H be a subgroup. Prove that p(E)…

A: Given: φ:G→H is a group homomorphism and E≤H. To prove: a) φ-1(E)≤G b) If E ⊲ H then φ-1E ⊲ G

Q: Let G be a group and H a normal subgroup of G. Show that if x,y EG Such that xyEH then 'yx€H-

A:

Q: 1. Let G₁ and G₂ be two groups, with identities e₁ and e2, respectively. Let G = G₁ x G2 and let H…

A:

Q: Let Φ: G to H be a group homomorphism such that Φ(g) ≠ eH for some non-identity element g in G,…

A: a) Let G and H be finite groups andϕ : G → H a homomorphism. Then|ϕ(G)| · |Ker (ϕ)| = |G|  (By…

Q: subgroup of G'. Prove that f(H) is a normal subgroup of G' and .Let f: G G' be an onto homomorphism…

A:

Q: b) Let M = 0(2) (O(n) is the orthogonal group) Calculate the tangent space TM and the normal space…

A: The set of matrices is a manifold. The tangent space to a manifold at a point a is the null space of…

Q: Let G be a group and N is a normal subgroup of G. Let f be a mapping from G to H/N defined b f(x) =…

A: Given :  G be a group and N is a normal subgroup of G. Let f be a mapping from G to H/N defined by…

Q: Question 3: a. Let (H,+) ≤ (R, +), let K = {2h : h E H}. Prove that (K,x) ≤ (R*,x). (Clarification:…

A: We shall use subgroup test to solve this problem

Q: 4. If 4: G → G' is a homomorphism, prove that ker þp is a normal subgroup of G.

A: To show that ker of a homomorphism is a normal subgroup of G

Q: How can I prove that f is one-to-one (injective)?

A: We know, a function/map f is one to one (injective) iff          For a ≠ b, f(a) ≠ f(b) for any two…

Q: Let G be a group. a) Define the center Z(G) b) Show that Z(G) is a subgroup of G c) Show that Z(G)…

A: WE HAVE GROUP G  THEN WE HAVE TO DEFINE CENTRE OF G AND SHOW THAT IT IS SUBGROUP AND NORMAL SUBGROUP…

Q: 5. True or False. Explain your answers. (a) Suppose f: G → H is a homomorphism and g € G. Then, the…

A: We have to answer that statements are true or false.

Q: Prove that if f is a continuous mapping of a set S (in E) into F and g is a continuous mapping of…

A: Given

Q: Let H and K be normal subgroups in G such that H n K = {1}. Show that hk = kh for all he H and k e…

A: