By Randall R. Holmes

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For centuries, the learn of elliptic curves has performed a crucial position in arithmetic. The prior century specifically has visible large development during this learn, from Mordell's theorem in 1922 to the paintings of Wiles and Taylor-Wiles in 1994. still, there stay many primary questions the place we don't even understand what kind of solutions to count on.

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We claim that (i1 , i2 )σ = σ , where σ = (i2 , i3 , . . , ir ). We have, (i1 , i2 )σ(i1 ) = i1 = σ (i1 ), (i1 , i2 )σ(ir ) = i2 = σ (ir ), (i1 , i2 )σ(ij ) = ij+1 = σ (ij ), (i1 , i2 )σ(k) = k = σ (k), 63 1 < j < r, k = i1 , i2 , . .

Since σ also sends 7 to 2, both sides of (*) send 7 to the same output as expected. 2 Theorem. Any element of Sn can be written as a product of disjoint cycles. Moreover, any two such factorizations are the same except possibly for the order of the factors (provided all cycles of length one are included ). 62 Proof. Let σ ∈ Sn . The algorithm illustrated above produces a product σ1 σ2 · · · σm of disjoint cycles. The proof that σ actually equals this product is Exercise 7–6. The proof of the uniqueness statement in the theorem is omitted.

Prove that Z(G) is a subgroup of G. 5–2 Fix n ∈ N. Let SLn (R) be the set of all n × n matrices over R having determinant 1: SLn (R) = {A ∈ Matn (R) | det(A) = 1}. Prove that SLn (R) is a subgroup of GLn (R) (= invertible n × n matrices over R). ) Hint: From linear algebra, we know that a square matrix is invertible if and only if its determinant is nonzero. Use the fact that det(AB) = det(A) det(B) for A, B ∈ Matn (R). 5–3 (a) Find the order of the element 9 in Z15 . (b) Find the order of the matrix A = 0 −1 in the group Mat2×2 (R).

### Abstract Algebra I by Randall R. Holmes

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