5. Let G be a finite group, and let class (₁), ..., class(xn) be all of the distinct non- singleton conjugacy classes in G. Show that |G| = |Z(G)| + n | class(x₁)| i=1 n = |Z(G)| + Σ|G : N(xi)|. i=1 This is called the class equation. (Hint: Let G act on X = G by conjugation, and use the Orbit Decomposition Theorem.)

Q: 3*. Let (F,+) be the group of polynomials over3 X {x, a2, a3,...} be the set of all positive integer…

A: To describe the subgroup generated by the positive powers of x in the additive group F[x]

Q: 3. Let G be a group. Recall that if x E G, we defined the conjugacy class of x in G, denoted by…

A:

Q: If you let G and H be two groups and consider the product group G x H and let K be the subset of G x…

A:

Q: Let G 0 -{(81%) = : a, b ER, a 0 (a) Show that G is a subgroup of SL(2). (b) For each of the…

A: Since you have posted a question with multiple subparts, we provide the solution only to the first…

Q: 7) Define a map ø1 from Z3 x Z → Z4 as follows: 01(m, n) = 4(m + n). Is ø1 a well-defined map? (If…

A: I just used the definitions of well definess and homomorphism

Q: 1) Letf: >→→ be defined by f(x) = 4x+1. (i) Define an operation that makes fan isomorphism of…

A:

Q: Let G and H be groups and consider the function f: G → G × H defined by f(g) = (g, eH ) where eH is…

A:

Q: Let G = {[1 0 0 1] ,[−1 0 0 1] ,[1 0 0 −1], [−1 0 0 −1]} . Is G ∼= K4? If yes, give an explicit…

A:

Q: Find the kernels and images of the following homomorphisms. Which of the homomorphisms are…

A:

Q: The class equation for a group G can be written |G| = |Z(G)| + E|CI(a), where the sum runs over…

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: 5*. Suppose that 0: G -+ H is a homomorphism. Prove that (a) (a")= (x)" for all positive integers n.…

A: To prove the properties of group homomorphisms

Q: Let G and G' be groups. (a) If p: G → G' is a function, what is required for p to be a homomorphism?…

A:

Q: Consider the group of nonzero real numbers, R* = R \ {0}, under mul- tiplication and the group of…

A:

Q: The class equation for a group G can be written |G| = |Z(G)| + 2|CI(a;)|, where the sum runs over…

A:

Q: Let G and H be finite groups. Prove the following statements. (a) If a: G H is a homomorphism, then…

A: Given that  and  are finite groups.We need to prove the following.(a) If  is a homomorphism, then…

Q: 1*. Let f G H be a group homomorphism. Prove: (a) If l is the identity of G then /(lg) is the…

A: To prove the required properties of a group homomorphism

Q: 5. Find the kernels and images of the following homomorphisms. Which of the homomorphisms are…

A:

Q: 5 Let G be the set of all functions f: R → R of the form f(x) = ax + b where a and b are real…

A: Introduction: Function composition is an operation where two functions are combined to form a new…

Q: Show that the groups (Z/4, +4) and (Z/5 – {[0]}, x5) are isomorphic.

A:

Q: Help: Let G and G' be groups with identity elements e and e' respectively and let f:G goes to G' be…

A: Let G and G' be groups with identity elements e and e' respectively and let f:G goes to G' be a…

Q: Find the kernels and images of the following homomorphisms. Which of the homomorphisms are…

A:

Q: 9. Let G and H be two groups, and G = {(9, en) | g = G}. Prove that G is a normal subgroup of G x H,…

A:

Q: A (ii), (iv) B E (i), (iii), (iv) All None (ii) Which of the following are true? (i): There is an…

A:

Q: If G and H are groups with identities 16 and ¹H, respectively, and f:G ---> H is a homomorphism,…

A:

Q: Explain why each of the following could not be the class equation of a group G. i) |G| = 3 + 3 + 3…

A:

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: 5. Let G be a group. Let H₁ CH₂ C H 3 C · of subgroups of G, and let H∞ = Un=1 Hn. be an infinitely…

A:

Q: 4. Let us define f : Z6 → Zž as f([r]) = [2ª]. Where [.] is the residue class interpreted correctly.…

A: