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

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However, what we cannot be sure of is that the definition is independent of the chosen representative. In an ideal world these two obvious ways of defining the product of cosets would coincide, solving both of our problems since the first definition is obviously independent of the chosen representative and the second one is obviously a coset. This leads to the notion of normal subgroups. 20 A subgroup U ≤ G of G is called normal or normal subgroup if for all g ∈ G and u ∈ U we have gug−1 ∈ U. (21) 49 We denote this by U ✂ G.

G. was a group, we may very well ask if there is a natural way to pass this structure on to the set of equivalence classes. 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.

39 such a subgroup has the form U = mZ for an integer m ≥ 0. e. m must lie between 1 and n and m is a divisor of n. We next want to compute the order of an element m ∈ Zn for positive integers m and n. For this we introduce the following notation. 43 For two integers a, b ∈ Z let lcm(a, b) := min{z > 0 | a and b divides z}, 0, if a, b = 0, if a = 0 or b = 0. 4. 44 Let m, n ∈ Z>0. Then o m = lcm(m, n) m is the order of m ∈ Zn. 12 takes the following form: o m = min k > 0 k · m = 0 = min k > 0 n divides k · m m · min{k > 0 | n divides k · m} m min m · k k > 0, n divides k · m = m min l > 0 n and m divide l = m lcm(m, n) .

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Algebraic Structures [Lecture notes] by Thomas Markwig Keilen


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