By Thomas Markwig Keilen
Read or Download Algebraic Structures [Lecture notes] PDF
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Extra resources for Algebraic Structures [Lecture notes]
Example text
Ak), (ak a1 . . ak−1), etc. all coincide. In order to avoid this ambiguity we recommend to start a cycle always with the smallest of the numbers a1, . . , ak. So far we have only introduced k-cycles for k ≥ 2. g. (1) or (3), and define each 1-cycle (a) (with a ∈ {1, . . , n}) to be the identity. 6 The permutations σ= 1 2 3 4 4 1 3 2 ∈ S4 and π= 1 2 3 4 5 4 1 3 2 5 ∈ S5 34 are all 3-cycles which permute the numbers 1, 4, 2 cyclically. In the above notation they are thus described as and σ = (1 4 2) π = (1 4 2).
63 If (G, ·) is a group and p = |G| a prime number then G is isomorphic to (Zp, +). 64 Let (G, ·) be a group, g ∈ G and n ∈ Z>0. Show there is a group homomorphism α : Zn −→ G with α 1 = g if and only if the order of g is a divisor of n. 65 Find all automorphisms of the group (Z10, +). 61 5 Check Digit Codes Nowadays products in shops all carry bar codes and are identified by them. Moreover, at the cash desk the bar code is scanned or typed in and that way you get charged the price. Sometimes the bar codes are not recognised correctly or the wrong number has been typed in.
1) · a13 or equivalently a1 + 3 · a2 + a3 + . . + a13 = 0. 63 We call these equations check digit equations. Does this still detect errors of Type I? Let’s go back to Equation (23) for this. The question finally comes down to checking whether ai = ai′ implies that ai − ai′ and 3 · (ai − ai′ ) are not equal to 0, which is the case since ai − ai′ is not divisible by 10 and thus also three times this number is not. Thus we are lucky. How about errors of Type II? If ai and ai+1 have been interchanged, then this comes down to the question whether 3 · ai + ai+1 = 3 · ai+1 + ai ⇔ 2 · (ai − ai+1) = 0 ⇔ 5 | ai − ai+1.