By Thomas Markwig Keilen

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

**Sample text**

In 51 days it is Wednesday. In order to get used to computations in Zn we recommend to do some computations with times and weekdays considering this new point of view. e. with a small number of elements, it is sensible to write down an addition respectively multiplication table, which shows for every two elements what their product is. In the case of Zn we get for n = 2, 3, 4 the following tables: + 0 1 0 0 1 1 1 0 + 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1 + 0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2 ✷ If for a normal subgroup N ✂ G the set of left cosets G/N is a group we may ask the question if the knowledge of the subgroups of G gives some information on the subgroups of G/N.

Mn} , for an arbitrary set M = {m1, . . , mn} of cardinality n, or if we simply study Sn = Sym {1, . . , n} . The two groups are isomorphic and we may thus identify them Sn is very important for practical reasons. In the lecture Grundlagen der Mathematik the group Sn will be used in connection with the determinant. Since the set {1, . . , n} is finite we can describe the permutation σ ∈ Sn simply by its value table. 1 Is σ ∈ Sn a permutation of the set {1, . . , n} then we can describe σ by the following scheme 1 2 ...

Concretely, if G is a group and U is a subgroup, is there a natural way to define a group operation on G/U? The natural should mean that the definition is somehow obvious. Given two cosets gU and hU we would want to define their product. There is of course a natural way to do this; both are subsets of G and we know already how to define the product of such subsets. What we do not know yet is if this product gives a coset again. If think for a second we might see another obvious way to define the product of gU and hU, namely as (gh) · U.

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

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