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

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

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Q: For any group G, Z(G) ≤ [G, G]. Prove or give a counter example

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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) =…

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Q: Let G be a group. Show that Z(G) = nEC(a). [This means the intersection of all subgroups of the form…

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Q: (a) Show that a group G is abelian, if (ab)² = a²b², for a, b € G[C.

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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: 2. Let G be a group, and let A be a non-empty subset of G. Define Ca(A) = {g G | gag¹ = a for all a…

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Q: If H is a finite subgroup of a group G and x ∈ G, prove that |N(H)| = |N(xHx-1)|.

A: Given: If H is a finite subgroup of a group G and x∈G. To Prove : NH=NxHx-1

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…

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Q: (14) Let G be a finite cyclic group of order n with identity element e. Prove that if a € G, then cd…

A: The order of every subgroup of a finite group G is divisor of the order of G.

Q: Suppose that the fundamental group of X is Z and p(xo) is finite. Find the fundamental group of X.

A: Given fundamental group of X is  ℤ and p-1(xo) = finite value Now we have to find the fundamental…

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 G be a group and a e G. Prove that C(a) is a subgroup of G. Furthermore, prove that Z(G) = NaeG…

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

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Q: Let H be a subgroup of a group G. Prove that the number of conjugates of H in G is |G:N(H)|.

A: It is given that H be a subgroup of a group G. It is required to prove that the number of conjugates…

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

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Q: Prove or give counterexample. For any group G, Z(G) ≤ [G, G].

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Q: a. Prove or Disprove. If H and K be normal subgroups of a group G and H is isomorphic to K, then G/H…

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Q: If H is the only subgroup of order o(H) in  a finite group G, show that H is the normal  subgrop of…

A: is the only subgroup of order in a finite group .We have to show that  is a normal subgroup of

Q: Let G be the group of integers under addition, and H the group of even integers under addition. Is G…

A: In group theory, two groups G and H are said to be isomorphic if there exists a bijective map ϕ:G→H…

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: f f is a homomorphism of group G into a  group G' with Kernal K, then K is a normal  subgroup of G.

A: If f is a homomorphism of group G into a group G' with Kernlal K, then we have to prove that K is a…

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 H be a subgroup of G and let K=⋂φ∈Aut(G)φ(H). Show that K is characteristic in G

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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…

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Q: Number

A: Let, G be finite group and order of group G is n and n is finite. Let, G' be a group such that  ϕ: G…

Q: 2. Let G be a group. Prove or disprove that Z= {x E G: xg= gx for all g€ G} Isa Subgroup of G.

A: To show Z is a subgroup of G, we need to show that (a) Z is non empty (b) For every a , b∈Z , we…

Q: Let G be a finite group and H1, H2, ., Hk be subgroups of G. (a) Show that k H1n H2n. n Hk < G. Hi…

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Q: Let G be a group and let H₁, H₂ be subgroups of G such that H₁ H₂ and H₂H₁. (a) Prove that the union…

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

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