By Mark Steinberger

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Every integer m > 1 has a unique prime decomposition: there is a factorization m = pr11 . . prkk , where pi is prime and ri > 0 for 1 ≤ i ≤ k, and pi < pj for i < j. Uniqueness means that if m = q1s1 . . qlsl with qi prime and si > 0 for 1 ≤ i ≤ l and qi < qj for i < j, then we must have k = l and have pi = qi and ri = si for 1 ≤ i ≤ k. CHAPTER 2. GROUPS: BASIC DEFINITIONS AND EXAMPLES 27 Proof The existence of a prime decomposition amounts to showing that any integer m > 1 is a product of primes.

Diﬀerent multiplications will in many cases give rise to isomorphic groups, but the number of isomorphism classes of groups with |X| elements must be ﬁnite. 5. Let n > 0. Then there are ﬁnitely many isomorphism classes of groups of order n. A countable union of ﬁnite sets is countable. 6. The set of isomorphism classes of ﬁnite groups is countable. 7. 1. Let X = {x, y} be a set with two elements. Which of the functions from X × X to X are group multiplications? Which are monoid multiplications?

7 37 The Group of Rotations of the Plane Here, we describe an important subgroup of Gl2 (R), the group, SO(2), of rotations of the plane. 1. For θ ∈ R, we deﬁne the rotation matrix Rθ by Rθ = cos θ sin θ − sin θ cos θ . Here, the sine and cosine are taken with respect to radian measure. Elements of M2 (R) act as transformations of the plane in the usual way. We identify the plane with the space of column vectors v = ( xy ). A matrix A = ac db acts on the plane by the usual multiplication of matrices on columns: A·v = ax + by cx + dy .

### Algebra [Lecture notes] by Mark Steinberger

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