Let Heis(F) be the Heisenberg group over the field F with the group operation being matrix multiplication. Show that if F is finite of order q then the order of Heis(F) is q3; conversely show that if F is infinite then Heis(F) is infinite.

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A: Consider the provided question, Your question is missing the last statement; I think you want to…

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Q: 1. Define x*y over R\ {-1}by x*y = x + y +xy. Prove that this structure forms an abelian group.

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A: Let d = gcd(m; n).We shall use induction on d.Assume  first that d = 1.

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A: Solution: We know that the permutaion τ on a non empty set A is a bijective map from A to A. That…

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A: Detailed explanation mentioned below

Q: be a Galois extension with Galois group G. Then F = KG.

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A: Here we have to prove that G is an abelian group.

Q: By applying Fundamental Theorem of group homomorphism, show that the quotient groups GL(n, R)/SL(n,…

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Q: 2. Let (G. .) be a group such that a.a = e for all a EG. Show that G is an abelian group.

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Q: Q3: (A) Prove that 1. There is no simple group of order 200.

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A: Let H be the set of all elements of the abelian group G that have finite order.  Prove that H is a…

Q: a) Prove that, the characteristic of an integral domain R is either zero or a prime number. b) Let G…

A: this question related to ring theory and group theory.

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A: Let G be a group of finite order n with identity e.           Since G is of finite order…

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A: Given: H≤Aut(k) To show: H is a subfield of K

Q: ii) Find the structure of its Galois group, G.

A: To Determine :- The structure of its Galois group, G.

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A: Suppose, G is a group of order p2 where p is prime.

Q: Let G be a group. Prove that (ab)1= a"b-1 for all a and b in G if and only if G is abelian

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Q: . Prove that the group Zm × Zn is cyclic and isomorphic to Zmn if and only if (m, n) = 1.

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Q: Let G be a finite non-abelian simple group and let q be prime, then [G] is

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Q: Prove that the symmetric group (S₂, 0) is abelian.

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A: Given that