Algebraic Structures [Lecture notes] by Thomas Markwig Keilen PDF

By Thomas Markwig Keilen

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Additional info for Algebraic Structures [Lecture notes]

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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. Type I: “Single Digit Errors” – i. e. just one digit is wrong. These are roughly 80% of the occuring errors. Type II: “Neighbour Transpositions” – i. e. two neighbouring digits have been interchanged.

Since disjoint cycles commute with each other we may assume without loss of generality 15 that k1 ≥ k2 ≥ . . ≥ kt. 15 “We may assume without loss of generality” in principle means that we only consider a special case and prove this one case. However, the other cases will follow in the same way. Doing only the 38 Moreover, since σ is not the neutral element id we know that k1 ≥ 2 and that the number s = max{r | 1 ≤ r ≤ t, kr ≥ 2} is defined. Thus σi = id for i = s + 1, . . , t and σ = σ1 ◦ . .

As mentioned, this leads to a lot of trouble for beginners — however, I would like to convince you that quotient groups are in principle something very simple. If we forget (like with the permutations in Sn) what the elements actually are, there remain only three simple rules for manipulating and computing with the elements of a quotient group. And all you need to remember, to be able to master quotient groups, are these rules! Like all groups quotient groups consist of a set together with a group operation.

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Algebraic Structures [Lecture notes] by Thomas Markwig Keilen


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