Let H and K be subgroups of a finite group G with H C KC G. Prove that |G:HI |G:K| |K:H].

Q: Consider a finite group G and H, K to be subgroups. Let H ⊂ K ⊂ G. Prove [G : H] = [G : K] * [K : H]

A:

Q: Let G1 and G2 be groups, and let G = G1 × G2. Write e and e2 for the identities of G1 and G2…

A:

Q: Suppose that G is an Abelian group and o is an automorphism of G. Prove that H = {xE G|6(x) = x-1}…

A:

Q: Let f: G H be an injective homomorphism of groups. Prove G is abelian if H is abelian.

A: Proof given below. Explanation:

Q: Let a be an element of a group G such that o(a) = r. Let m be a positive integer such that (m,r) =…

A: a∈GOa=rgcdm,r=1 We have to prove Oam=r

Q: If f is a homomorphism of a group G into a group G' with Kernal K. Let a € 6 be such that f(a) = a'…

A: We are given that f : G → G' is a homomorphism. Kernel of f is K.     => for all elements x in K,…

Q: Let G be a group and let a be an element of G. Prove that a func- tion fa : G → G defined by fa(g) =…

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 A = {fm,b : R → R | m ± 0 and fm,b(x) : = mx + b, m, b e R} be the group of affine m b functions…

A: We are given two groups.  A={fm,b:ℝ→ℝ|m≠0 and fm,b(x)=mx+b,m,b∈ℝ} B=mb01 | m,b∈ℝ,m≠0. We need to…

Q: Let G be a finite group of order n and identity element 1. a. Show that g" =1 for any g e G. b. An…

A:

Q: Let G be a finite group and H1, H2,…., Hk be subgroups of G. .... (a) Show that N H; = Hị n H2 n..n…

A:

Q: Let G be a group and let x ∈ G. Prove that if a has order of 12, then x3 has order 4.

A:

Q: Let G be an abelian group and m be a positive integer. Then the set mG = {mn 1 x E G} and G(m) = {x…

A: Given : Let G be an abelian group and m be a  positive integer.To Show :  the set mG={mn∣x∈G} and…

Q: (c) Show that the Euclidean group E(2) is generated by the reflec- tions. More precisely, show that…

A: The Euclidean group E2 is generated by the reflections. That is any isometry of ℝ2 is a product of…

Q: Let (G1, +) and (G2, +) be two subgroups of (R, +) so that Z+ ⊆ G1 ∩ G2. If φ : G1 → G2 is a group…

A:

Q: 1, and let ø denote Let m and n be positive integers with gcd(m, n) Euler's o-function (Definition…

A: Given: Let's determine the following a) Let's verify that (a,b) is a generator for the group ℤnℤ×ℤmℤ…

Q: (2) Let G be a finite group and H and K be subgroups of G such that HCKCG. Prove that [G: H] = [G:…

A: Let us prove this beautiful result using Lagrange's theorem. Lagrange's theorem = If G is a finite…

Q: Prove that H= { |ne Z} is a subgroup of GL2(R) under multiplication.

A:

Q: Let G be a finite group. Let xeG, and let i>0. Then prove that o(x) gcd(i,0(x))

A: To prove the required identity on the order of the group element

Q: Let G = Z[i] = {a+bi | a, b € Z} be the Gaussian integers, which form a group under addition. Let y…

A:

Q: Let : G → H be a group homomorphism. Show that is a subgroup of H. Im(6) := {p(g) | g = G}

A: Let G,∘ and H,* be two group and ϕ:G→H be a homomorphism. Let ϕG is a non-empty subset of H. Since…

Q: Let G be a finite group and let G’ be a group. Let ø : G → G’ be a homomorphism.  Prove that | ø[G…

A:

Q: Suppose that G is a finite group and let H and K be subgroups of G. Prove that |HK| = |H||K|/|HN K|.

A:

Q: Let H be a subgroup of G and N(H)={g ∈G:gh =hg for all h ∈ H}.Prove N(H) is a subgroup of G.

A:

Q: Let : G H be a group homomorphism. Prove that Kery = {e} if and only if is injective.

A: