100A: Abstract Algebra ... Well-ordering principle for natural numbers (Beachy
and Blair Axiom 1.1.2). • Division Algorithm (Beachy and Blair Theorem 1.1.3).
100A: Abstract Algebra Some1 key definitions and theorems Definitions • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
Associative binary operation An identity (with respect to some binary operation) An inverse of an element (with respect to a binary operation and a given identity element) Group (G, ∗) Subgroup Subgroup diagram of a group Abelian group Finite group vs. infinite group The order of a group, the order of a subgroup The cyclic subgroup generated by an element a ∈ G, a Generator of a group Cyclic group The order of an element of a group, o(a) for a ∈ G (finite order vs. infinite order) The subgroup generated by a subset S of a group Given G a group and subsets S, T ⊆ G, the product ST The direct product of two groups G1 × G2 For integers: multiples, divisors Greatest common divisor gcd(a, b) = (a, b) Lowest common multiple lcm(a, b) = [a, b] Relatively prime integers, prime numbers, composite numbers Congruence modulo n (for n ∈ N=0 ) Congruence class of a ∈ Z modulo n ∈ N=0 ; representatives of congruence classes Zero divisor (in Zn ), unit (of Zn ) Euler’s ϕ-function Domain, codomain, and image of a function Composition of functions Onto (surjective); one-to-one (injective); bijective Identity function Inverse of a function Permutation of a set Cycle of length k (Disjoint) cycle decomposition of a permutation Transposition Even permutations vs. odd permutations Permutation group Group isomorphism, isomorphic groups Rigid motion Group homomorphism Kernel of a homomorphism Normal subgroup Factor group of a group with respect to a homomorphism or with respect to a normal subgroup. Left coset of a subgroup in a group determined by an element of the group. Similarly, right coset. • Index of a subgroup in a group, [G : H]. 1
This list may not be exhaustive: it is intended as a starting point only.
Important examples • • • • • • • • • •
Z and mZ for m ∈ Z Zn , the additive group of integers modulo n Z× n , the multiplicative group of units modulo n Sym(S), the symmetric group on S Sn , the symmetric group of degree n Dn , the dihedral group of degree n An , the alternating group on n elements Mn (R) GLn (R) SLn (R)
Some named theorems • • • • • • • • • •
Well-ordering principle for natural numbers (Beachy and Blair Axiom 1.1.2) Division Algorithm (Beachy and Blair Theorem 1.1.3) Fundamental Theorem of Arithmetic (Beachy and Blair Theorem 1.2.7) Euler’s Theorem (Beachy and Blair Theorem 1.4.11) Fermat’s Little Theorem (Beachy and Blair Corollary 1.4.12) Lagrange’s theorem (Beachy and Blair Theorem 3.2.10) Cayley’s theorem (Beachy and Blair Theorem 3.6.2) Fundamental Homomorphism Theorem (Beachy and Blair Theorem 3.8.9) First Isomorphism Theorem (Beachy and Blair Theorem 7.1.1) Second Isomorphism Theorem (Beachy and Blair Theorem 7.1.2)
