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Download PDF by Thomas Markwig Keilen: Algebraic Structures [Lecture notes]

By Thomas Markwig Keilen

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Extra resources for Algebraic Structures [Lecture notes]

Example text

However, we are willing to pay this price for a representation which is both shorter and better organised. Moreover, in applications we usually know very well what the set of definition of a permutation in question is, and anyhow the really important information is which numbers are move. ✷ The representation of a permutation by the cycle scheme would not be very helpful if it only applied to permutations which are indeed cycles, while all other permutations would have to be represented by the clumsier two-row scheme.

6, but instead of the symbol “∼” it is common to write x ≡ y (n) or x ≡ y (mod n) in order to indicate that x is congruent to y modulo n. ✷ We want to close this section with an important theorem, the Theorem of Lagrange. Its main statement is that the order of a subgroup divides the order of the group itself. The following lemma is a central building block of the proof of Lagrange’s Theorem. 9 Let G be a group, U ≤ G and g ∈ G. Then the map lg : U −→ gU : u → gu is a bijection. In particular, all left cosets of U in G have the same cardinality |U|.

Ak−1 ak b1 . . bn−k a2 a3 . . ak a1 b1 . . bn−k ∈ Sn, then we call σ a k-cycle, and we say that it cyclically permutes the numbers a1, . . , ak. Such a map can be represented in a much more efficient way by the one-line scheme: σ = (a1 . . 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.

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Algebraic Structures [Lecture notes] by Thomas Markwig Keilen

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