prove: Let H be a subgroup of a group G. Then |g1H| = |g2H| for any g1, g2 ∈ G. In particular |H| = |gH| for any g ∈ G. Additionally, |Hg1| = |Hg2| and |H| = |Hg| for any g, g1, g2 ∈ G.

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

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Q: Let H and K be a subgroups of a group G. Define the relation as follows: ab if and only if a = hbk,…

A: Let e be the identity element of G.  As H and K are subgroups of G,          e ∈ H and e ∈ K.  Now,…

Q: Let G be a group, let H < G, and let x E G. We use the notation xHx1 to denote the set of elements…

A: In a group G, two element g and h are called conjugate when h = x g x-1 for some x ∈ G For an…

Q: Q2: Let (G,) be a commutative group, and let the set H consist of all elements of G with finite…

A: Given a group G and a set H of G with the given conditions. We need to show that H is a normal…

Q: ts] Let G be a group, and let H be a subgroup of G. Show that if G is cyclic then H is also cyclic.…

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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.     =&gt; for all elements x in K,…

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…

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Q: let G be a group and H a subgroup of G. prove that for any element gEG holds that gH=H if and only…

A: We can solve the given question as follows:

Q: 7. Let G be an infinite cyclic group. Give an example of a subgroup H in G such that H {e} or G. 8.…

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Q: Let G be a group and let g, h ∈ G. Show that | gh | = | hg |. Remember that | a | denotes the order…

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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: 48. Let G be a group and g = G. Show that Z(G) = {x ≤ G : gx = xg for all g = G} is a subgroup of G.…

A: A non empty subset S of a group (G,∗) is said to be a subgroup of G if S itself a group. That is i)…

Q: Let G and H be groups. Let p : G → H be a homomorphism and let E < H be a subgroup. Prove that p(E)…

A: Given: φ:G→H is a group homomorphism and E≤H. To prove: a) φ-1(E)≤G b) If E ⊲ H then φ-1E ⊲ G

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

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

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Q: Let G be a group and suppose that G only has finitely many distinct subgroups. Show that |G| < ∞.

A: If possible  let G is of infinite  order .In particular let  G = {a1,a2,...,an,...} .Let us consider…

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: suitte (b) Given two groups (G,) and (H, *). Suppose that is a homomorphism of G onto H. For BH and…

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Q: Suppose the class equation of a group G is |G| = 1 + 4 + 5 + 5 + 5. Show that G has an element of…

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Q: Let G be a finite group. Let x E G and let G be the conjugacy class of r. Prove that |#C| < |[G,…

A: Given G be a finite group. Let x∈G. Define: xG=gxg-1: g∈G Let G:G=m. Define a map π:G→GG:G defined…

Q: 3. Let G be a finite group of order |G| = n. Prove that G is isomorphic to a subgroup of Sym(G)…

A: The objective is to prove that G is isomorphic to a subgroup of symmetric group.

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: 3) Let G and H be two groups with e and e' as identities. Define P = G x H by P = G x H = { (g,h) |…

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

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

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Q: Let G be a group which is an algebraic structure.Let H and K be the subgroups of G with H⊂K⊂G. Let…

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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: (4) Let G be a group. Prove that Z(G) is a normal subgroup of G.

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

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