Group Theory

Group theory is a branch of abstract algebra concerned with groups: sets equipped with an operation satisfying precise rules for composition, identity, and inverses. Groups provide a common language for symmetry, whether their elements are numbers, permutations, matrices, or geometric transformations. The subject investigates their internal structure, their relationships, and their actions on other objects, connecting algebra with geometry and many other areas of mathematics. (math.mit.edu)

Definition and basic examples

A group consists of a set $G$ and a binary operation, usually written as multiplication. Its defining axioms are:

• Closure: $ab$ belongs to $G$ whenever $a,b\in G$.
• Associativity: $(ab)c=a(bc)$.
• Identity: an element $e$ satisfies $ea=ae=a$.
• Inverses: each $a$ has an element $a^{-1}$ satisfying $aa^{-1}=a^{-1}a=e$.

Closure can instead be incorporated into the definition of the operation as a map $G\times G\to G$. The identity and each element’s inverse are unique. Commutativity is not required; a group satisfying $ab=ba$ for all elements is an abelian group. (math.mit.edu)

The integers form a group under addition, with identity $0$ and inverse $-a$. In modular arithmetic, the residue classes modulo a positive integer $n$ form an additive group with $n$ elements. Both examples are cyclic groups, meaning that repeated application of the operation to one generator produces every element. Every cyclic group is abelian, although not every abelian group is cyclic. (ocw.mit.edu)

The symmetric group $S_n$ consists of all permutations of $n$ objects, composed as functions, and has $n!$ elements. The dihedral group of a regular $n$-gon contains its $n$ rotations and $n$ reflections. Invertible square matrices of a fixed size over a field form a group under matrix multiplication. These examples demonstrate that the operation need not resemble ordinary numerical multiplication. (jmilne.org)

Subgroups, quotients, and structural maps

A subgroup is a subset that itself forms a group under the inherited operation. Its left cosets, $gH=\{gh:h\in H\}$, partition the original group. For a finite group, this partition gives Lagrange’s theorem:

where $[G:H]$ is the number of left cosets. Consequently, a subgroup’s order divides the group’s order. (math.mit.edu)

A normal subgroup $N$ satisfies $gNg^{-1}=N$ for every $g\in G$. Its cosets form a quotient group $G/N$, with multiplication $(gN)(hN)=ghN$. Normality ensures that this operation does not depend on the representatives chosen. (math.mit.edu)

A group homomorphism is a map $\varphi:G\to K$ preserving multiplication: $\varphi(ab)=\varphi(a)\varphi(b)$. Its kernel consists of elements mapped to the identity and is normal. A bijective homomorphism is an isomorphism, identifying groups with the same algebraic structure despite different descriptions. The first isomorphism theorem states that $G/\ker\varphi$ is isomorphic to the image of $\varphi$. (jmilne.org)

Actions and finite-group structure

A group action assigns transformations of a set $X$ to group elements so that $e\cdot x=x$ and $(gh)\cdot x=g\cdot(h\cdot x)$. An element’s orbit comprises all points reachable from it; its stabilizer comprises the group elements fixing it. For finite groups, the orbit–stabilizer formula relates these quantities:

Actions connect abstract groups to concrete symmetries and support counting arguments. (jmilne.org)

The Sylow theorems constrain finite groups through their prime-power subgroups. If $|G|=p^am$, with $p$ prime and $p\nmid m$, subgroups of order $p^a$ exist. They are mutually conjugate, and their number divides $m$ and is congruent to $1$ modulo $p$. Such restrictions often establish whether a subgroup must be normal. (crypto.stanford.edu)

A simple group is nontrivial and has no proper nontrivial normal subgroup. A solvable group admits a finite series of subgroups, each normal in the next, whose successive quotients are abelian. These concepts describe different ways in which groups resist or permit decomposition. (math.mit.edu)

Representations and continuous symmetry

Representation theory studies homomorphisms from groups to groups of invertible linear transformations of a vector space. It translates group structure into linear algebra. An irreducible representation has no nonzero proper invariant subspace; decomposing representations into irreducible components reveals how symmetries act on different parts of a system. (math.mit.edu)

A Lie group is both a group and a smooth manifold, with smooth multiplication and inversion. Examples include rotation groups and invertible real or complex matrix groups. Its associated Lie algebra describes infinitesimal structure near the identity. This provides tools for studying continuous symmetries alongside the discrete symmetries of finite groups. (ocw.mit.edu)

Connections and applications

In Galois theory, groups of field automorphisms encode relationships among polynomial roots. Over a field of characteristic zero, a polynomial is solvable by radicals exactly when its Galois group is solvable. This explains why no radical formula solves every polynomial equation of degree five. (jmilne.org)

In crystallography, groups organize rotations, reflections, and translations preserving crystal structures. Three-dimensional periodic structures have 230 crystallographic space-group types. In quantum mechanics, representations describe how states transform under symmetry operations; Lie groups provide the mathematical framework for continuous symmetries in physical theories. (iucr.org)