Diskrete Strukturen [Lecture notes] by Manuel Bodirsky

By Manuel Bodirsky

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Wir nennen die Eckpunkte des Quadrats a, b, c, d; hierbei ist a oben links, und die u ¨brigen Elemente folgen gegen den Uhrzeigersinn. Diese Gruppe hat den Namen D4 , eine sogenannte Diedergruppe. ). Die folgende Tabelle zeigt f¨ ur jedes Element sein Inverses an. x id ↔ x−1 id ↔ ↔ ↔ ↔ id id id ↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔ id ↔ ↔ ↔ id ↔ ↔ id ↔ ↔ ↔ id id ↔ ↔ ◦ id ↔ ↔ Und schließlich die Kompositionstabelle. ↔ 35 id id ↔ ↔ ↔ Bedeutung identische Abbildung Drehung um 90 Grad Drehung um 180 Grad Drehung um 270 Grad Vertikale Spiegelung Horizontale Spiegelung Spiegelung mit Achse bd Spiegelung mit Achse ac Eckenpermutation a a a b a c a d a d a b a c a a bcd bcd b c d cda b c d da b b cd ab c bcd cba b c d ad c b c d bad b cd dc b Abbildung 10: Symmetrien eines Quadrates mit den Ecken a, b, c, d.

Also ist Y = A ∩ B. Da |Y | ≥ k − 1 ≥ 1, widerspricht dies der Annahme, dass A und B disjunkt sind. Betrachten wir also zuletzt die Situation, dass |Ya | > k oder |Yb | > k. Dann ist |Y | ≥ k. Nach Induktionsannahme gibt es dann k disjunkte A-B Pfade sogar in G , also auch in G. Korollar 54. Sei G ein endlicher Graph und a, b ∈ V (G) so dass {a, b} ∈ / E(G). Dann ist die maximale Anzahl paarweise unabh¨ angiger Pfade von a nach b gleich der minimalen Anzahl von Knoten aus V (G) \ {a, b}, die a von b in G trennt.

Also gilt a ≡ b (mod q1 q2 ). Zur¨ uck zu unserer Behauptung. Die Primzahlen p und q sind teilfremd, also gen¨ ugt es d d nach dem eben bewiesenen Lemma zu zeigen, dass m ≡ c (mod p) und m ≡ c (mod q). Falls p ein Teiler ist von m, dann gilt cd = m1+hφ(n) ≡ 0 ≡ m (mod p). Ansonsten gilt cd = m1+hφ(n) (wie oben) (p−1)(q−1)h =m·m (Proposition 25) ≡ m · (1)(q−1)h (mod p) (kleiner Fermat; p teilt nicht m) ≡ m (mod p) Analog zeigen wir, dass cd ≡ m (mod q), und mit Lemma 37 erhalten wir die Behauptung.

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