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: Let G be a group. Suppose G=Z6xZ4. State all subgroups of G. Categorized(classify) the…

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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: Show that a group G is abelian if and only if the mapping f: G->G, given by f(x) = x2, is a…

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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: If H and K are subgroups of G and g belongs to G, show that g(H N K) = gH n gK.

A: Given: H and K are subgroups of G and g.To prove: gH∩K=gH∩gK

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: Determine the group Z42/ and find a group that it is isomorphic to it. Generalize your observation.

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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: 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: 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: 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: et f be an isomorphism from a grou onto a Group H. et a be an element in G and the orde Fa is 5. it…

A: We have to choose the correct answer from the given option according to the given condition

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: Prove the next statement: If H is a subgroup of a finite abelian group G, then G has a group…

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

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Q: Show that the quotient group Q/Z is isomorphic to the direct sum of prufer group

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Q: Show that the multiplicative group Z is isomorphic to the group Z2 X Z2 8,

A: We know that if two groups are isomorphic than they have same number of elements i.e. their…

Q: Let G be a group and let A1, A2, B1, B2 be the proper nontrivial subgroups of G. Suppose that A1×A2…

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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.      => for all elements x in…

Q: Let F and H be subgroups of group G and let FCH. Prove (G : F) = (G : H)(H : F).

A: It is given that F and H are subgroups of G and F⊂H.

Q: Let (G, ) be a group. Define a new binary operation * on G by the formula a * b = b · a for all a, b…

A: We proved (G,*) is a group if it satisfied the following axioms.