Abstract Algebra I

An abelian (or commutative) group is a group in which the operation commutes: a·b = b·a for all elements a, b. Named after Niels Henrik Abel for his work on the solvability of polynomial equations, abelian groups are structurally much simpler than general groups and are completely classified: the Fundamental Theorem of Finite Abelian Groups says every finite abelian group is a direct product of cyclic groups of prime-power order, unique up to reordering. Familiar examples include the integers under addition, ℤₙ, and vector spaces under addition.

The dihedral group Dₙ (order 2n) is the group of symmetries of a regular n-gon, consisting of n rotations and n reflections. It is the smallest interesting family of non-abelian groups (for n ≥ 3) and provides the standard first example encountered when moving beyond cyclic and abelian groups. Dₙ is generated by a rotation r of order n and a reflection s of order 2, subject to the single relation srs = r⁻¹, which encodes that reflecting then rotating is the same as rotating the opposite way then reflecting.

The direct product G × H of two groups G and H is the set of ordered pairs (g, h) with componentwise operation, itself forming a new group whose order is |G|×|H|. Direct products are the simplest way to build larger groups from smaller ones and, via the Fundamental Theorem of Finite Abelian Groups, every finite abelian group decomposes as a direct product of cyclic groups of prime-power order. The internal direct product characterizes when a group G actually splits as (is isomorphic to) A × B using two of its own normal subgroups.

A group action is a formal way of describing how a group G 'acts' on a set X by permuting its elements, compatibly with the group's operation. Actions turn abstract group elements into concrete symmetries — rotations acting on points, permutations acting on a set, or a group acting on itself by conjugation — and are the main tool for connecting abstract group structure to combinatorial counting via the orbit-stabilizer theorem. This machinery underlies the proof of the Sylow theorems and Burnside's counting lemma.

The isomorphism theorems are a set of foundational results, largely due to Emmy Noether, that describe precisely how quotient groups, subgroups, and homomorphisms relate to each other. The First Isomorphism Theorem says every homomorphism factors as an embedding of G/ker(φ) into the codomain, identifying the quotient with the image. The Second and Third build on this to relate subgroups of a quotient back to subgroups of the original group. Together they are the fundamental toolkit for understanding group structure by comparing a group to its quotients and images.

Lagrange's theorem states that for any finite group G and any subgroup H of G, the order of H divides the order of G, with the quotient |G|/|H| equal to the index [G:H], the number of distinct cosets of H. The proof is a clean counting argument: the cosets of H partition G into equal-sized blocks. Named after Joseph-Louis Lagrange, who proved a special case for permutation groups before the group concept was formalized, this theorem is one of the most heavily used facts in finite group theory — it instantly rules out many possible subgroup orders.

A normal subgroup is a subgroup N of a group G that is invariant under conjugation by every element of G: gNg⁻¹ = N for all g ∈ G. Normal subgroups are exactly the subgroups whose left and right cosets coincide, which is precisely the condition needed to make the set of cosets G/N into a group under a well-defined operation. They were introduced by Évariste Galois in his study of when polynomial equations are solvable by radicals, and every kernel of a group homomorphism is a normal subgroup, and conversely.

Given a group G and a normal subgroup N ⊴ G, the quotient group G/N is the set of cosets of N in G, made into a group by the operation (aN)(bN) = (ab)N. It is well-defined precisely because N is normal, and it formalizes the idea of 'collapsing' N to a single identity element while retaining a coherent group structure on what remains. Quotient groups are central to the first isomorphism theorem, which identifies G/ker(φ) with the image of any homomorphism φ.

The Sylow theorems, proved by Peter Ludwig Mejdell Sylow in 1872, are a partial converse to Lagrange's theorem: while Lagrange only says subgroup orders must divide |G|, the Sylow theorems guarantee that subgroups of certain sizes actually exist, and describe how many there are. Specifically, they concern Sylow p-subgroups — subgroups whose order is the largest power of a prime p dividing |G|. These theorems are the single most powerful tool for classifying finite groups of a given order and are proved using group actions on sets of subsets or on the set of Sylow subgroups by conjugation.

The symmetric group Sₙ is the group of all bijections (permutations) of a set of n elements under composition, with order n!. Every finite group embeds into some symmetric group (Cayley's theorem), making Sₙ in a precise sense the universal finite group. Sₙ is generated by transpositions, splits into even and odd permutations via the sign homomorphism, and its subgroup of even permutations, the alternating group Aₙ, is simple for n ≥ 5 — the fact underlying Galois's proof that the general quintic has no solution by radicals.

A presentation describes a group compactly by naming a set of generators and a set of defining relations (equations among the generators) that, together with the axioms of a group, determine the group completely: ⟨generators | relations⟩. Rather than listing every element and every product, a presentation gives a recipe — start with the free group on the generators, then quotient by the smallest normal subgroup containing the relators, so that the relations hold and nothing extra is forced. Presentations let you specify enormous or infinite groups with a handful of symbols, and they underlie combinatorial group theory, computational group theory (e.g. the Todd–Coxeter algorithm), and topology (fundamental groups of spaces are naturally presented this way).

The free group F(S) on a set S is the 'most unconstrained' group generated by S: its elements are reduced words (strings of generators and their inverses with no adjacent cancellations), and no relations hold among the generators except those forced by the group axioms themselves. Every group generated by S is a quotient of F(S) — this is exactly what makes presentations ⟨S | R⟩ = F(S)/N(R) work, since any relations you want to impose are added on top of the free group as the baseline. Free groups are infinite whenever |S| ≥ 1 — even F({a}) with a single generator is isomorphic to the infinite cyclic group ℤ — and F(S) for |S| ≥ 2 is a rich source of non-abelian, torsion-free examples used throughout combinatorial and geometric group theory.

Two elements a and b of a group G are conjugate if b = gag⁻¹ for some g ∈ G, and this relation partitions G into disjoint conjugacy classes — sets of elements that are 'the same up to a change of viewpoint.' Conjugacy classes are central to understanding a group's structure: a subgroup is normal exactly when it is a union of conjugacy classes, the class equation constrains possible group orders (it's the key tool in proving nontrivial centers for p-groups), and in the symmetric group Sₙ, conjugacy classes correspond exactly to cycle types, making them fully computable by counting cycle-shape partitions of n.

The center Z(G) of a group G is the set of elements that commute with every element of G — the 'universally agreeable' elements that behave the same no matter what they're combined with. Z(G) is always a normal (indeed abelian) subgroup of G, and it measures how far G is from being abelian: Z(G) = G exactly when G itself is abelian, while Z(G) = {e} means no nontrivial element commutes with everything. The center also equals the union of all singleton conjugacy classes, linking it directly to the class equation and to deep structural results such as every nontrivial finite p-group having a nontrivial center.

A semidirect product N ⋊ H generalizes the direct product by letting H act on N nontrivially via a homomorphism φ: H → Aut(N), so that combining elements twists N's component according to H's action instead of leaving it alone. Where the direct product N × H requires both factors to be normal and to commute with each other, the semidirect product only requires N to be normal, and it captures exactly the structure of groups G that split as G = NH with N ⊴ G, H ≤ G, N ∩ H = {e} — precisely the situation that arises for dihedral groups, and more generally whenever a group has a normal subgroup with a complement.

A group is a set equipped with a single operation that combines two elements to produce a third, satisfying four properties: closure, associativity, an identity element, and inverses. Groups are the simplest and most fundamental algebraic structure — the integers under addition, non-zero rationals under multiplication, and the symmetries of a shape are all groups, despite looking completely different on the surface.

A cyclic group is a group that can be generated by repeatedly applying a single element and its inverse. Cyclic groups are the simplest possible groups structurally, and every subgroup of a cyclic group is itself cyclic, making them a natural base case for classification theorems.

A group homomorphism is a structure-preserving map between two groups: it sends products to products, so the algebraic relationships in the source group are mirrored exactly in the target group. Homomorphisms let us compare groups, transfer theorems between them, and formally define when two groups are 'the same' (isomorphic).

A permutation group is a group whose elements are permutations (bijections) of a set, with the group operation being function composition. The symmetric group Sₙ, consisting of all permutations of n objects, is the archetypal example, and by Cayley's theorem every group can be realized as a permutation group.

A subgroup is a subset of a group that is itself a group under the same operation. Subgroups let us study a large group by decomposing it into smaller, self-contained pieces, and they are the objects that Lagrange's theorem and the classification of group structure revolve around.