# Algebra and number theory, U Glasgow notes by Baker.

By Baker.

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19, we have the prime power factorization n = pr11 pr22 · · · prt t , where for each j, pj is a prime, 1 rj and 2 p1 < p2 < · · · < pt . We set 0 if any rj > 1, µ(n) = µ(pr11 pr22 · · · prt t ) = t (−1) if all rj = 1. So for example, if n = p is a prime, µ(p) = −1, while µ(p2 ) = 0. Also, µ(60) = µ(22 × 3 × 5) = 0. 2. The M¨ obius function µ is multiplicative. Proof. This follows from the definition and the fact that the prime power factorizations of two coprime natural numbers m, n have no common prime factors.

For a finite set X, the unique n ∈ N0 for which there is a bijection n −→ X is called the cardinality of X, denoted |X|. If X is infinite then we sometimes write |X| = ∞, while if X is finite we write |X| < ∞. 3 without proof to give some important facts about cardinalities of finite sets. 5 (Pigeonhole Principle). Let X, Y be two finite sets. a) If there is an injection X −→ Y then |X| |Y |. b) If there is a surjection X −→ Y then |X| |Y |. c) If there is a bijection X −→ Y then |X| = |Y |. The name Pigeonhole Principle comes from the use of this when distributing m letters into n pigeonholes.

A B C), (A C B)): these give rotations and can only fix a colouring that has all sides the same colour, hence | FixG (σ)| = 4. , σ = (A B), (A C), (B C)): each of these gives a reflection in a line through a vertex and the midpoint of the opposite edge. For example, (A B) fixes C and interchanges the edges AC, BC, it will therefore fix any colouring that has these edges the same colour. There are 4 × 4 = 16 of these, so | FixG ((A B))| = 16. Similarly for the other 2-cycles. By the Burnside formula, 1 120 number of distinguishable colourings = (64 + 2 × 4 + 3 × 16) = = 20.