Group Theory lec • Basic Structures: – Semigroup: A set with an associative binary operation. Examples include (N, +). – Monoid: A semigroup with an identity element. Examples include (N0 , +). – Group: A monoid where every element has an inverse. Examples include (Z, +) and (R∗ , ×). • Subgroups: – A subset H ≤ G is a subgroup if: ∗ H is non-empty. ∗ H is closed under the group operation. ∗ H is closed under taking inverses. – Normal subgroup: N ⊴ G if gN g −1 = N for all g ∈ G. Examples include the center Z(G). – Cosets: For H ≤ G, left cosets are gH = {gh | h ∈ H}. • G Quotient map N Subgroup inclusion {e} • Homomorphisms: – – – – A map ϕ : G → H is a homomorphism if ϕ(ab) = ϕ(a)ϕ(b). Kernel: ker(ϕ) = {g ∈ G | ϕ(g) = eH }. Image: Im(ϕ) = {ϕ(g) | g ∈ G}. Isomorphism: A bijective homomorphism, e.g., Z6 ∼ = Z2 × Z3 . • Quotient Groups: – G/N is the set of cosets {gN | g ∈ G} with the operation (gN )(hN ) = (gh)N . – G/N forms a group if N ⊴ G. Example: Z/nZ. • Commutators and Derived Subgroups: – Commutator: [a, b] = aba−1 b−1 measures non-commutativity. – Derived subgroup: G′ = ⟨[a, b] | a, b ∈ G⟩ is the smallest normal • Group Actions: – A group G acts on a set X if there is a map G×X → X satisfying: ∗ g · (h · x) = (gh) · x for all g, h ∈ G and x ∈ X. ∗ e · x = x for all x ∈ X. – Orbit: Orb(x) = {g · x | g ∈ G}. Example: Action of S3 on {1, 2, 3}. – Stabilizer: Stab(x) = {g ∈ G | g · x = x}. Example: Identity stabilizes every element. • Sylow Theorems: – If |G| = pk m with p ∤ m, then: ∗ G has a Sylow p-subgroup of order pk . ∗ All Sylow p-subgroups are conjugate. ∗ The number of Sylow p-subgroups satisfies np ≡ 1 (mod p) and np | m. – Applications: Prove S3 has a normal Sylow 3-subgroup. • Nilpotent Groups: – A group is nilpotent if it has a central series: G = G0 ≥ G1 ≥ · · · ≥ Gn = {e}, where Gi+1 ⊴ Gi and Gi /Gi+1 ≤ Z(G/Gi+1 ). – Examples: Z is nilpotent; S3 is not. • Category Theory: – A category consists of objects, morphisms, and composition laws. – Products: X × Y satisfies projection properties. – Limits: Generalize products for arbitrary diagrams. – Example: Groups form a category with homomorphisms as morphisms. 2