(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: Prove that a group G is abelian if and only if the mapping f: G→G, given by f(x)=x-1,  is a…

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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…

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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 C(x) represent the centralizes of a. If x is an element of group G with the order of x be 5 (|x|…

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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: A homomorphism f of a group G into a group G' is an ismorphism iff ker f = {e}.

A: We have to proof that a homomorphism f of a group G into a group G' is an isomorphism if and only if…

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: Let G be a group and N is a normal subgroup of G. Let f be a mapping from G to H/N defined by f(x) =…

A: Given : Let G be a group and N is a normal subgroup of G and Let f be a mapping from G to H/N…

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

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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.

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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: 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 finite group and let H be a normal Sylow p-subgroupof G. Show that a(H) = H for all…

A: Let α be an automorphism of G, and let p be a prime such that H is a Sylow p-subgroup of G.

Q: Let G and H be groups, and let N be a normal subgroup of G. Decide whether each of the following two…

A: (a) The statement "The group  is abelian if and only if G and H are both abelian" is…

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

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Q: A homomorphism f of a group G into a group  G' is an ismorphism iff ker f={e}.

A: We have to proof that a homomorphism f of a group G into a group G' is an isomorphism if and only if…

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: (B) Let G be a set of all real numbers except 0. Define on G by a b = |a|b, where lal is the…

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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: Let G be a group. The centre of G is defined as the set Z(G) = {g ∈ G : gx = xg for all x ∈ G}.…

A: Let G be a group. The center ZG of group G is defined as follows. ZG = g∈G : gx = xg , ∀x∈G We need…

Q: (3) For any group G, Z(G) ≤ [G, G].

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

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: Let (G,*) be a group such that x² = e for all x E G. Show that (G,*) is abelian. (Here x² means x *…

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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: 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…