Let φ : G → H be a group homomorphism. (a) Prove that Ker(φ) is a normal subgroup of G. (a) Prove that Im(φ) is a subgroup of G. Is it normal? When?

Q: Recall that the center of a group G is the set {x € G | xg = gx for all g e G}. Prove that he center…

A:

Q: Prove that, if H is a subgroup of a cyclic group G, then the quotient group G/H is also cyclic.

A:

Q: Show that if H and K are subgroups of an abelian group G, then {hk|h € H and k e K} is a subgroup of…

A: A set G is called a group if it satisfies four properties  Closure property:  ab∈G  where a,b ∈G…

Q: Let G be a nonabelian group. If H and K are cyclic subgroups of G, does it follow that H ∩ k is also…

A: Given that G is non abelian group. Given that H and K are cyclic subgroups of G. The cyclic group…

Q: 2. Let G be a group. Prove or disprove that z= { XE G: xg = gx for all ge G} is a Subgroup of G.

A:

Q: Prove that H x {1} and {1} x K are normal subgroups of H x K, that these subgroups general H x K,…

A:

Q: 6. If G is a group and H is a subgroup of index 2 in G; then prove that H is a normal subgroup of G:

A: I have proved the definition of normal subgroup

Q: Let G be a group and g E G. Show that Ø:Z→G by Ø(n) = g" is a homomorphism and isomorphism.

A:

Q: (2) of order 5 is in H. Let G be a group of order 100 that has a subgroup H of order 25. Prove that…

A:

Q: (a) of G'. Show that if y :G → G' is a group homomorphism then Im(y) is a subgroup

A: According to the given information, For part (a) it is required to show that:

Q: Prove that if H is cyclic, then ø[H] is also cyclic.

A:

Q: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Let G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G

Q: Compute the Cayley table and find a familiar isomorphic group of G/N where G is a group and N is any…

A:

Q: Suppose that 0: G G 5a group homomorphism. Show that 0 $(e) = 0(e) (ii) For every geG, (0(g))= 0(g)*…

A:

Q: Let ø be a homomorphism from a group G to a group H. Let K be a subgroup of H. Prove that ø (K) = {…

A:

Q: Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

A: Given that Let G be a group and H be a normal subgroup of G. If H and G/H are solvable then so is G.

Q: . Let H be a subgroup of a group G. Prove that the set HZG) = {hz | h E H, z E Z(G)} is a subgroup…

A:

Q: H be a subgroup of G.

A: We have to find out the truth value of the given statements. It is given that H is a subgroup of G.…

Q: If a cyclic group T of G is normal in G; then show t subgroup of T is a normal subgroup in G

A: Given: A cyclic group T of G is normal in G.

Q: KE Syl-(G). Prove that (a). HG and KG. (U). G has a cyclic subgroup of order 77. Syl(G),

A:

Q: Question 10. Prove in detail that G(T) is a subgroup of G.

A: Here we use one step subgroup test

Q: . Let G be the additive group Rx R and H = {(x,x) : x E R} be a subgroup of G. Give a geometric…

A:

Q: Let G be a group, prove that the center Z(G) of a group G is a normal subgroup of G.

A: Let G be a group. Consider the subgroup ZG=x∈G | ax=xa.

Q: . Let H and K be normal subgroups of a group G such nat HCK, show that K/H is a normal subgroup of…

A:

Q: Question 2: If p is a homomorphism of group G onto & with kernel K and N is a normal subgroup of G.…

A: Introduction: If there exists a bijective map θ:G→G' for two given groups G and G', then θ is…

Q: Let G be a finite group. Let E G and let xG be the conjugacy class of x. Prove that x| < |[G, G]],…

A:

Q: I need help with attached abstract algebra question to understand it.

A: To show that the subset H of G is indeed a subgroup of G

Q: Let G be a group and H a subgroup of G. If [G: H] = 2 then H ⊲ G, where [G: H] represents the index…

A:

Q: Let (G, -) be an abelian group with identity element e Let H = {a E G| a · a · a·a = e} Prove that H…

A: To show H is subgroup of G, we have show identity, closure and inverse property for H.

Q: Let G be a group and let Z(G) be the center of G. Then the factor group G/Z(G) is isomorphic to the…

A: We have to check G/Z(G) is isomorphic to group of all inner automorphism of G or not. Where, Z(G) is…

Q: QUESTION7 Let g be a fixed element of a group G.Prove that H = {x EG: gx = xg } is a subgroup of G.

A:

Q: Let G =U(9) and H= (8). Explain why H is a normal subgroup of and construct the group table for the…

A:

Q: et H < G. Then Gµ = {g € G | gHg¬ = H} is a subgroup of G that has H has a -ormal subgroup. %3D

A:

Q: Let H be a subgroup of G of index 2. Prove that H is a normal sub-group of G.

A: the prove is given below...

Q: Let G be a group acting on a set S. Let x and y be elements in S such is in the orbit of x under the…

A:

Q: 2. Let G be a group and let H be a subgroup of G. Define N(H) = { x = G | xHx™¹ = H}. Prove that…

A:

Q: Let C be a normal subgroup of the group A and let D be a normal subgroup of the group B. Prove that…

A:

Q: Let G  be a group and H a normal subgroup of G. Show that if x,y in G such that xy in H then yx in H

A: We are given that H is a subgroup of G. ⇒) Assume H is a normal subgroup of G. So,…

Q: Let G be a group with identity element e, and let H and K be subgroups of G. Assume that (i) H and K…

A:

Q: Suppose that 0: G G is a group homomorphism. Show that 0 $(e) = ¢(e') (1) For every gEG,…

A:

Q: 7. Let G be a group, prove that the center Z(G) of a group G is a normal subgroup of G.

A:

Q: '. Assume that G is a group such that for all x E G, x * x = e. Prove that G is an abelian group.

A: Consider any two elements a and b in G. So, a,b,ab,ba∈G. Note that I am directly writing the…

Q: If H is a subgroup of G such that [G : H] = 2, then show that H is a normal subgroup of G.

A: Suppose H≤G such that [G:H] = 2. Thus H has two left cosets (and two right cosets) in G.

Q: Prove that if G is a group of order 60 with no non-trivial normal subgroups, then G has no subgroup…

A:

Q: In the group (Z, +), find (-1), the cyclic subgroup generated by -1. Let G be an abelian group, and…

A:

Q: Let G be a non-trivial group. Prove that Aut(G) × Aut(G) is Aut(G x G). a proper subgroup of

A:

Q: Let G be a group and D = {(x, x) | x E G}. Prove D is a subgroup of G.

A:

Q: 3. Let (G, *) be a group and let H and K be subgroups of G. Prove or disprove each of the following…

A:

Q: Let o be an automorphism of a group G. Prove that H = {x E G | $(x) = x} is a subgroup of G.

A: One step subgroup test: Let G,∘ be a group, then H,∘ is a subgroup of G,∘ if and only if, 1) H is…