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Ak). (15) b. A 2 − cycle is also called a transposition. A transposition τ = (i j) is thus a permutation where only the two numbers i and j are exchanged while all the others are fixed. c. ,n}, and we will simply denote it by id. 5 The interpretation of the notation in Equation (15) is obvious, the first element a1 is mapped to the second element a2, the second one is mapped to the third one, and so on, while the last one, namely ak, is mapped to the first one a1. This closes the cycle. Note here that the cycles (a1 .

14 every permutation is a product of transpositions of consecutive integers. 40 Let σ1 = τ˜1 ◦ · · · ◦ τ˜r and σ2 = τ˜r+1 ◦ · · · ◦ τ˜r+s be given as products of such transpositions of consecutive numbers. By induction on r + s we see that sgn(σ1 ◦ σ2) = (−1)r+s = (−1)r · (−1)s = sgn(σ1) · sgn(σ2). , and b. follows by induction on k. For c. let σ = τ1 ◦ · · · ◦ τk = τ1′ ◦ · · · ◦ τl′ with transpositions τi, τj′ ∈ Sn. Then by b. (−1)k = sgn(σ) = (−1)l, and thus either k and l are both even or both odd.

C) Quotient Group We have now gathered all results which we need in order to formulate the theorem on the quotient group. In the hope that the notation g for the left coset gU of a subgroup will make it easier to concentrate on the calculations with the elements of the quotient group by simply clouding the fact that the element g = gU is actually a set, we will adopt this notation right away. 36 Let (G, ·) be a group and U ✂ G be a normal subgroup of G. Then21 g · h = g · h, for g, h ∈ G/U. (22) With this multiplication as binary operation the set G/U is a group.

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Algebraic Structures [Lecture notes] by Thomas Markwig Keilen

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