Group Theory

What is symmetry, and what are all the ways a structure can be transformed into itself?

5 chapters4 min read5 turning points 1 open problem

Branched from: Galois Theory + Non-Euclidean Geometry
Branched into: Abstract Algebra + Representation Theory
Figures: Arthur Cayley, Ludwig Sylow, Walter Feit, John Thompson, Pyotr Novikov, Sergei Adian, Daniel Gorenstein, Robert Griess, Michael Aschbacher

In brief

Group theory is the mathematics of symmetry. A group is a collection of transformations, such as the rotations of a cube, the shuffles of a deck or the symmetries of an equation's roots, that can be combined and undone. The same abstract structure turns up in equations, geometry, crystals, molecules and particle physics.

It emerged in the nineteenth century from three directions at once: Galois's symmetries of equations, Klein's idea that a geometry is defined by its group of motions, and number theory. Its greatest achievement is the classification of all finite simple groups, the "atoms" of finite symmetry, a proof spread over tens of thousands of pages that ends with a giant called the Monster.

Key ideas

GroupEnters 1854: A set with an operation that is associative, has an identity, and has inverses: the rules any collection of symmetries automatically satisfies.
Lagrange's theoremEnters 1770 – 1771: The size of a subgroup always divides the size of the group. A group of 15 elements can have subgroups of size 1, 3, 5 or 15, and no others.
Simple groupEnters 1955 – 2004: A group that cannot be broken into smaller pieces. Every finite group is built from simple groups, the way molecules are built from atoms.
Sporadic groups and the MonsterEnters 1955 – 2004: Twenty-six finite simple groups that fit no infinite family. The largest, the Monster, has about $8 \times 10^{53}$ elements.
Group presentationEnters 1968: Describing a group by generators and the relations they satisfy. Simple-looking presentations can hide infinite or unknowable groups, as the Burnside problem showed.

Chapter I

Three Roots

Groups arrived from several directions at once. In Galois theory they were the symmetries of an equation's roots. In geometry, Felix Klein's Erlangen programme of 1872, on the non-Euclidean geometry page, defined each geometry by its group of motions. In number theory, Gauss had worked with structures that were groups in all but name. What united them was an abstraction. In 1854 Arthur Cayley defined a group simply as a set with a multiplication obeying a few rules, whatever the elements happen to be. For decades almost no one noticed.

The abstraction paid off. Ludwig Sylow, teaching at a Norwegian secondary school, proved in 1872 that every finite group contains certain predictable subgroups. Theorems about all groups at once, whatever their origin, became possible.

Chapter II

Atoms of Symmetry

Every finite group can be broken down into simple groups, which cannot be broken down further, much as molecules break into atoms. The grand question became: what are all the finite simple groups? Some come in infinite families, such as the alternating groups, the even shuffles of $n$ objects. Others, discovered one by one from 1861 onward, seemed to fit nowhere.

In 1963 Walter Feit and John Thompson proved that every finite simple group other than the cyclic ones has even order. It was a single theorem whose proof filled an entire issue of a journal. It showed the classification might be possible, and a coordinated campaign followed, led by Daniel Gorenstein. It found eighteen infinite families and exactly twenty-six "sporadic" exceptions. The largest, the Monster, with about $8 \times 10^{53}$ elements, was constructed by Robert Griess in 1982.

Chapter III

A Proof Too Large to Read

The classification was announced complete in 1983. Then a gap surfaced: one case rested on an unfinished, unpublished manuscript. Michael Aschbacher and Stephen Smith closed it in 2004 with over 1,200 pages. The whole proof is spread over hundreds of papers by more than a hundred authors, and a streamlined "second-generation" version is still in progress. Whether any single person can check it is a real question, and one reason machine-checked proofs matter.

Chapter IV

A Closer Look: The Symmetries of a Square

Cut a square out of paper and label its corners 1, 2, 3, 4 clockwise. How many ways can you pick it up and put it back so that it fills the same hole? There are four rotations (by $0°$ , $90°$ , $180°$ , $270°$ ) and four flips (about the two diagonals and the two lines through the midpoints of opposite sides). That makes eight symmetries in all, and they form a group, called $D_4$ .

Combine two of them: $r$ , a quarter-turn clockwise, and $s$ , a flip about the vertical axis. Doing $r$ then $s$ does not give the same result as doing $s$ then $r$ . Try it with the labelled square: the corners end up in different places. In fact $s$ followed by $r$ equals $r$ turned the other way followed by $s$ . The group is non-commutative: order matters, just as it does for turning a Rubik's cube or rotating a book first about one axis and then another.

Lagrange's theorem predicts which smaller groups can sit inside $D_4$ : their sizes must divide 8. They do. There is the group of just "do nothing" (size 1), a single flip with "do nothing" (size 2), the four rotations (size 4), and a few others of sizes 2 and 4. None has size 3, 5, 6 or 7.

Abstraction is the point. The group $D_4$ is also the Galois group of the equation $x^4 - 2 = 0$ , whose four roots form a square in the complex plane. The same eight-element structure describes a square of paper and the symmetries of a polynomial's roots. Group theory studies the structure once and applies it everywhere.

Chapter V

Groups That Refuse to Be Finite

Not all questions about groups end in classification. In 1902 William Burnside asked whether a group generated by finitely many elements, each of which cycles back to the identity after a fixed number of steps, must be finite. In 1968 Pyotr Novikov and Sergei Adian showed it need not be. For exponent 5 and two generators, the question is still open.

The study of how groups act on vector spaces, as matrices, became representation theory. There the Monster, of all things, turned out to be connected to number theory and string physics.

Turning points5

1854REFORMULATION
Cayley defines the abstract group
Arthur Cayley defines a group abstractly, as a set of symbols with a multiplication table satisfying certain rules, rather than as a particular set of permutations. The same structure can then be recognised in matrices, quaternions and geometric motions. The idea took decades to catch on.
Arthur Cayley
› Sources (1)
- Cayley, A. (1854). On the theory of groups, as depending on the symbolic equation θⁿ = 1. Philosophical Magazine 7(42): 40–47.
1872PROOF
The Sylow theorems
Ludwig Sylow, a Norwegian schoolteacher, proves that every finite group contains subgroups whose sizes are the largest prime powers dividing its order, and describes how many there are. They are the first powerful tools for dissecting an arbitrary finite group, and still the starting point of the structure theory.
Ludwig Sylow
› Sources (1)
- Sylow, L. (1872). Théorèmes sur les groupes de substitutions. Mathematische Annalen 5: 584–594.
1963PROOF
The odd order theorem
Walter Feit and John Thompson prove Burnside's 1911 conjecture that every group of odd order is solvable, so every non-abelian finite simple group has even order. The 255-page proof, filling an entire issue of a journal, showed that a full classification of finite simple groups might be possible, and set its methods.
Walter Feit, John Thompson
› Sources (1)
- Feit, W. & Thompson, J. G. (1963). Solvability of groups of odd order. Pacific Journal of Mathematics 13(3): 775–1029.
1968DISPROOF
The Burnside problem has a negative answer
William Burnside asked in 1902 whether a group generated by finitely many elements, each of which returns to the identity after $n$ steps, must be finite. Pyotr Novikov and Sergei Adian show that for large odd $n$ the answer is no: such groups can be infinite. Efim Zelmanov's solution of the "restricted" version earned a Fields Medal in 1994.
Pyotr Novikov, Sergei Adian
› Sources (1)
- Novikov, P. S. & Adian, S. I. (1968). On infinite periodic groups I–III. Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 32: 212–244, 251–524, 709–731.
1955 – 2004PROOFContested
The classification of finite simple groups
Dozens of mathematicians, organised in large part by Daniel Gorenstein, prove that every finite simple group belongs to one of eighteen infinite families or is one of twenty-six sporadic exceptions. The largest exception, the Monster, was constructed by Robert Griess in 1982. The proof runs to tens of thousands of journal pages, and a "second-generation" streamlined version is still being written.
What is disputed
The completion date is disputed. The classification was announced as complete in 1983, but a major part, the "quasithin" case, turned out to rest on an unpublished and incomplete manuscript. Michael Aschbacher and Stephen Smith filled the gap with a 1,221-page work published in 2004. Some mathematicians still question whether a proof of this size can be fully certain, which motivates the ongoing second-generation proof.
Daniel Gorenstein, Robert Griess, Michael Aschbacher
› Sources (2)
- Solomon, R. (2001). A brief history of the classification of the finite simple groups. Bulletin of the AMS 38(3): 315–352.
- Aschbacher, M. & Smith, S. D. (2004). The Classification of Quasithin Groups (2 vols.). American Mathematical Society.

Applications

Where it is used

Chemistry
Molecular symmetry and spectroscopy
Each molecule has a symmetry group, and group theory predicts which of its vibrations absorb infrared light, which electronic transitions are allowed, and how orbitals combine. It is standard equipment in physical chemistry.
› Sources (1)
- Cotton, F. A. (1990). Chemical Applications of Group Theory (3rd ed.). Wiley.
Puzzles
Rubik's cube: God's number is 20
The positions of Rubik's cube form a group of about $4.3 \times 10^{19}$ elements. In 2010, using group theory and donated computer time, a team proved that every position can be solved in at most 20 moves.
› Sources (1)
- Rokicki, T., Kociemba, H., Davidson, M. & Dethridge, J. (2014). The diameter of the Rubik's Cube group is twenty. SIAM Review 56(4): 645–670.
Virology↗ Biology · Virology
Why so many viruses are icosahedra
A virus must build a closed shell from many copies of one protein, which is a problem in the symmetry groups of the sphere: the icosahedral group is the largest finite rotation group, so it allows a container to be assembled from the greatest number of identical subunits in identical environments. Caspar and Klug's quasi-equivalence theory of 1962 enumerates the permitted shells by a triangulation number, and the resulting list matches the capsids observed.
› Sources (2)
- Caspar, D. L. D. & Klug, A. (1962). Physical principles in the construction of regular viruses. Cold Spring Harbor Symposia on Quantitative Biology 27: 1–24.
- Twarock, R. & Luque, A. (2019). Structural puzzles in virology solved with an overarching icosahedral design principle. Nature Communications 10: 4414.

Open problems

Where the map runs out

Open

Is the Burnside group B(2,5) finite?

Open as of 2026.

The free Burnside group $B(2, 5)$ is the most general group generated by two elements in which every element returns to the identity after 5 steps. For exponents 2, 3, 4 and 6 such groups are finite, and for large odd exponents they can be infinite. Whether $B(2,5)$ is finite is unknown.

Why it is hard: Exponent 5 is too small for the Novikov–Adian machinery, which needs large exponents, and too large for the direct computations that settled exponents up to 4 and 6. Its largest finite quotient is known (Zelmanov's theorem guarantees one), but not whether the group itself is that quotient.
What resolving it unlocks: It would locate precisely where finiteness breaks down in one of the oldest questions of group theory.

› Sources (1)

Vaughan-Lee, M. (1993). The Restricted Burnside Problem (2nd ed.). Oxford University Press.