By Thomas Markwig Keilen
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Additional info for Algebraic Structures [Lecture notes]
41. 46 Find all subgroups of (Z33, +). 47 Consider for m, n, a, b ∈ Z>0 the element am, bn ∈ Zm × Zn in the group (Zm × Zn, +). Show that the order of this element can be computed as follows o am, bn = lcm o am , o bn = lcm lcm(a, m) lcm(b, n) , a b and that it is a divisor of lcm(m, n). In particular, the group Zm × Zn is not cyclic if m and n share a common divisor apart from one.. 48 Compute the order of 621, 933 ∈ Z21 × Z33. 49 Let σ and π be two disjoint cycles in Sn of length k respectively l.
5 One left coset of U in G is always known. e. the subgroup itself is always a left coset. Moreover, one should note that the possible representatives of a left coset are precisely the elements in that coset. In particular, uU = U for each u ∈ U. ✷ The possibly most important example in our lecture is the set Z/nZ of the left cosets of the subgroup nZ in the group (Z, +). In order to be able to describe all left cosets in this example and in order to give for each of them a most simple representative we need the principle of division with remainder for integers.
E) Which weights wi would have been suitable in the check digit equation in order not to loose the property that errors of Type I are detected? The important point was that ai = ai′ i. e. that the map ⇒ ωi · ai = ωi · ai′ , µωi : Z/10Z → Z/10Z : a → ωi · a is injective, and hence bijective since Z/10Z is a finite set. In other words, µωi is a permutation of the set Z/10Z. This leads to the following generalisation and definition. 1 Let (G, ·) be a group, g0 ∈ G a fixed element, and let π1, . .
Algebraic Structures [Lecture notes] by Thomas Markwig Keilen