By Thomas Markwig Keilen

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**Extra resources for Algebraic Structures [Lecture notes]**

**Example text**

Mathematicians are lazy people, or maybe one should better say that they are efficient. Therefore, they have thought of a way to represent a permutation in such a way that each of the numbers 1 to n has to be written at most once, not twice. For this we need the notion of a cycle, which permutes k of the numbers 1, . . , n cyclically. 4 a. Let {1, . . , n} = {a1, . . , ak} ∪ {b1, . . , bn−k}, k ≥ 2, and σ= a1 a2 . . 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, .

In view of the definition of α this means that necessarily α(2) = α(6) must hold true. This, however, is correct since 2 and 6 differ i·π by 4 and e 2 ·4 = 1. 20) sgn : Sn −→ {−1, 1}. 50 implies in particular |Sn/An| = |{−1, 1}| = 2. 11 the following equation: |An| = n! 2 The following isomorphism theorems are easy applications of the above Homomorphism Theorem. 53 (1st Isomorphism Theorem) If G is a group, U ≤ G and N ✂ G. Then ∼ UN/N. U/U ∩ N = Proof: We leave the proof to the reader as an exercise.

9} → Z/10Z : a → a = a + 10Z = {a + 10z | z ∈ Z}, i. e. identifying the digit with the residue class represented by the number. Viewing the digits in the article number as elements of Z/10Z that way, the check digit becomes just the sum of the “digits”. E. g. 0 + 1 + 3 + 4 + 1 + 2 + 5 + 4 + 7 + 1 + 8 + 0 = 36 = 6. C) Does this allow to detect errors? Otherwise it is of no use. Certainly we will not be able to detect all errors, thus we have to distinguish certain types of errors! Some statistics tell us that the following two types are the most common ones.

### Algebraic Structures [Lecture notes] by Thomas Markwig Keilen

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