Let Φ: G to H be a group homomorphism such that Φ(g) ≠ eH for some non-identity element g in G, where eH is the identity in group H. Furthermore, suppose that G has order 11 and that H has order 22. a) What are the possible orders of Φ(G)? Explain. b) Is Φ a one-to-one map? Explain. c) Prove that Φ(G) is a normal subgroup of H. Show your work.

Q: on G by a * b = ba. Prove that G is isomorphic

A:

Q: Defn: Let  H be a subgroup of G.  NG(H)={g∈ G | gHg-1=H } is called the normalized of H in G. a)…

A: First, we define a Normal subgroup.A subgroup H of a group G is said to be normal subgroup of G if…

Q: Let : G H be a homomorphism. Let (G) = {(g) | ge G}. Show that (G) is a subgroup of H.

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: 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: (3) Let H be a subgroup of a group G. Let a E G. Prove that a¹Ha = {a¹ha he H} is a subgroup of G.

A:

Q: Prove that any non-trivial subgroup of (Z, +) is isomorphic to (Z, +).

A:

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 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 a e G. Prove that $(a") = ¢(a)" for all n e Z.

A:

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: 1. Let G be a group. Define the center of G by C(G) = {x € G: xg = Prove that C(G) is always a…

A:

Q: ¹) Let H be a subgroup of a group G. Define NĠ (H) : = {g €G | g Hg¯¹ = H}. Prove that N(H) is a…

A:

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: (True/False) Suppose G is a group. Then : G x G →G, (x,y) → ay is a group homomorphism. (True/False)…

A:

Q: Let G be a group with the operation * and let H be a subgroup of G. Show that if H and G/H have the…

A: To show that H is a normal subgroup of G, we need to demonstrate that for every element g in G, the…