Let f be a homomorphism of a group G onto a group G' and H = Ker. f, K' is any normal subgroup of G' and K = {x ¤ G : f(x) ≤ K′} = ƒ−¹ (K′). then K is a normal subgroup of G containing Hand G/K = G′/ K'.

Q: 1. Let M2(R) = {( ) la, b, c, d eR and ad bc + 0}. You already know that M2(R) forms a group under…

A: We have to prove that H is subgroup of M2(R) :

Q: 2. For both parts below, let p :G → G' be a group homomorphism. Let H' be a subgroup of G'. Show…

A:

Q: 15. Let (G, *) be a group, and let H₁, H₂,..., Hk be normal subgroups of G such that H₁ H₂0... Hk =…

A:

Q: Let H be a normal subgroup of a finite group G and let x E G. If gcd(lxl, IG/H\) = 1, show that x E…

A: Let H be a normal subgroup of a finite group G and x∈G, Given: gcdx,GH=1. To Show: x∈H.

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: a b - Let H = 5, d E R, ad 0 Is H a normal sub- group of GL(2, R)?

A: Given: H=ab0d  a, b, d∈ℝ, ad≠0. We have to check whether H is a normal subgroup of GL2, ℝ or not.

Q: 6. Let : (G,*) → (G',') be a group isomorphism, and let a € G. Prove that (a). G is abelian if and…

A:

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: 2. Let G be the group of 2 x 2 matrices under addition and H = {(ab): a +d=0}. Prove that H is a…

A:

Q: If f: G→G' be a homomorphism of groups and e, e’ be the identities in G and G′ respectively Then (i)…

A: If f:G→G' be a homomorphism of groups and e, e' be the identities in G and G' respectively. Then we…

Q: Let G be a group with operation and H be a group with operation f: G H is a mapping that has…

A:

Q: Consider the groups (R,+) and (R x R. +). Define the map Rx (RX R) → Rx R defined by ra (z. y) =…

A: The given groups are ℝ, + and ℝ×ℝ, +. The map ↦: ℝ×ℝ×ℝ→ℝ×ℝ is defined as, r↦x, y=x+ry, y Definition:…

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: 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: 2. Let G = GL(2, R). Prove that the following two subsets of GL(2, R) are subgroups of GL(2, R). (a)…

A: 2a Given set is A=a00d:a>0 and d>0. We have prove that set A is the subgroup of GL2,R. For the…

Q: 4. Let G be a group and G' be the subgroup generated by the set S = {x¹y¹xy x, y = G}. G' is called…

A: Let G be a group, G' be its commutator and N be a normal subgroup of G such that G/N is abelian.…

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: 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: unctions are well-defined group homomorphisms? Se elections; you cannot score less than 0 on this…

A:

Q: 1. Let K be a subgroup of R*. Let H = {g E GL(n, R): det (g) E K}. Prove that H is a subgroup of…

A:

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: Let H and K be subgroups of a group G. (a) Define HK = {hk | he H, ke K}. Show that if K is normal…

A: We will solve all the three parts. Given that  H and  K are subgroup of G

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 be the group of all transformations on R which have the form x → ax + b where a, b E R, a ± 0.…

A: Let us assume f(x)=ax+b a,b ∈ℝ  and G is the collection of all such f.

Q: Suppose 0: G-→ G be a homomorphism of groups. Show that: (i) Kere is normal in G (ii) The mappping…

A: Normal subgroup: A subgroup N of a group G is said to be a normal subgroup of G if  gNg-1=N , for…

Q: 1. Let (G, *) be a finite group of order n. Then (a). Let n = pq (p, q are a prime numbers) and let…

A: This is a problem of Group Theory.

Q: Let M N be normal subgroups of a group G an of G such that [N, H] ≤ M and [M, H] = 1. Prove t and x…

A: Given that M⩽N and N is normal subgroup of G and His a subgroup of G such that [N,H]⩽M and…

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: 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: Show that S = SU(2) contains a subgroup isomorphic to S'.

A: Let's define S1 as the set { (x,y)∣ x2+y2 = 1 } ⊂ R2 we may think of S3 as S3={ (a,b) ∈ C2:…

Q: Let G be a group. Define Z(G) = {x G | xg = gr for all g € G}. Prove Z(G) ≤ G. Find Z(S3) and Z(D4).

A: Solution:Let  G be a group. Define Clearly, eSuppose be  arbitrary.Then,  [since, x,y]Thus, 2nd…

Q: (a) Consider the symmetric group Sx on the set X = R-{0, 1}. Let f be the function defined by the…

A:

Q: (a). If G is an abelian group, then (b). If G an abelian group, and group. (c). If o(a) = n, then…

A: This is a problem of Group Theory.

Q: Find the distance of the point (1,3) from the line 2x-3y+9=0 measured       along a line x-y+1=0.

A:

Q: 11.1.12. Let G be a group, and assume that H and K are normal subgroups of G with trivial…

A: Given below solution Explanation:Step 1: Step 2: Step 3: Step 4:

Q: Given that f be a homomorphism of a group G onto a group G' and H= Ker(f), K' is any normal subgroup…

A: Given that f be a homomorphism of a group G onto a groupG' and H=Ker(f), K' is any normal subgroup…

Q: 4. Given G and G' be two groups with respect to multiplication operation and 0: G→ G' is a…

A:

Q: x = {[8 -d]|a,b€R} Prove that X*: =X\ {[%]} is a subgroup of GL(2, R).

A: