Let H be a non-empty subset of a group G. Then H is a  subgroup of G iff      a,b∈H⇒ab\power{-1}∈H, where b\power{-1} is the inverse of b in G.

Q: Show that for any subgroup of a group the inverses of the  elements of a left coset form a right…

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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: 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: 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 H and K be finite subgroups of a group G and a E G. Then prove that |HaK| = |H||K| /|HnaKa-|.

A: Given that H and K are the finite subgroups of a group G and also an element a such that a∈G Here,…

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: Chapter 2, exercise 10.2(a): Let H and K be subgroups of a group G. Prove that the intersection…

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Q: 5. Define the right regular action of a group G on itself. Show it is a group action. Is it…

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Q: Let us consider that S is subset of group G such that                      S = { x in G ; x^2=e }…

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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 H be a proper subgroup of a finite group G. Show that G is notthe union of all conjugates of H.

A: Let H be a proper subgroup of a finite group G. We how prove that G is not the union of all…

Q: Let N be a normal subgroup of a finite group G.  Prove that the order of the group element gN in G/N…

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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: Assume that G is an abelian group. Let H C G be a subgroup. Prove that the factor group G/H is…

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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: Let G, and G, be two groups. Let H and H, be normal subgroups of G G, respectively then @ H, x H, 4G…

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Q: Exhibit a Sylow 2-subgroup of S4. Describe an isomorphism fromthis group to D4.

A: It is given that a Sylow 2-subgroup of S4. So, we have, S4=4!=24=3×23. let k be a Sylow 2-subgroup…

Q: Let G be a group and H, K≤ G normal subgroups of G. Prove HnK≤G.

A: Given that  is a group and  are normal subgroups of .We need to prove that  is a normal subgroup.We…

Q: Suppose that A is subset of a group G such that              A = {a : a2 =e  , a in G }    Where e…

A: Sol:- The subset A consists of all elements a in the group G such that a squared is equal to the…

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: 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: Give an example of a finite group G with two normal subgroups H and K such that G/H = G/K but H 7 K.

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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: G

A: According to the question, Let's start by defining the dihedral groups D3 and D5,  So, D3 is the…

Q: Let G be a group with the operation * and let H be a subgroup of G. Show that if H and G/H have the…

A: To show that H is a normal subgroup of G, we need to demonstrate that for every element g in G, the…

Q: Give an example of a group G and subgroup H ≤ G such that ♬ is not normal in G.

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Q: Show that every subgroup H of the group G of index two is normal.

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Q: 19. Let H and K be subgroups of a group G. Prove that gH ngK is a coset of HOK in G.

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Q: For any group G, GIZ(G) is isomorphic to Inn(G)

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Q: Consider a group G with operation * and identity element e. Show that for any element a in G, there…

A: In any group G, we define a group as a set G equipped with an operation * that combines any two…