Advanced Groups Summary

Abstract Algebra · Axiom Academy

Let's review how permutation groups, group actions, Sylow theory, and abelian group classification reveal deep structural properties of groups. Symmetric Group: Sₙ consists of all permutations of n elements. Order is n! and it's the universal group for all groups of order n. Cycle Notation: Write permutations as products of disjoint cycles. (1 3 5)(2 4) sends 1→3→5→1 and 2→4→2. Order Formula: Order of a permutation equals the LCM of its cycle lengths. Disjoint cycles commute. Alternating Group: Aₙ contains even permutations (even number of transpositions). Order is n!/2 for n ≥ 2. Cayley's Theorem: Every group G embeds in S|G| via the regular action, proving all groups are "permutation groups." Definition: A group G acts on set X if there's a map G × X → X satisfying e·x = x and g·(h·x) = (gh)·x . Orbit: The orbit of x is Orb(x) = g·x | g ∈ G , the set of elements reachable from x. Stabilizer: The stabilizer of x is Stab(x) = g ∈ G | g·x = x , elements fixing x. Always a subgroup. Examples: Conjugation, left multiplication, matrix groups on vector spaces, symmetry groups on geometric objects. Setup: For group G with |G| = p n m where p is prime and gcd(p,m) = 1, a Sylow p-subgroup has order p n . Existence: At least one Sylow p-subgroup always exists for each prime dividing |G|. Conjugacy: All Sylow p-subgroups are conjugate to each other, hence isomorphic. Counting: If nₚ is the number of Sylow p-subgroups, then nₚ ≡ 1 (mod p) and nₚ | m .

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