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A prison essay became group theory
The unsolved quintic
For hundreds of years nobody answered whether degree-five equations have a formula like the ones Italian mathematicians had found for degrees three and four. Lagrange noticed that solving these polynomials is related to how certain algebraic expressions stay symmetric when their variables are permuted. He also floated the opposite question: could a formula be proven impossible? He never solved the problem, and at the time it was almost a side, hobbyist question, since most people cared more about applications to physics.
Abel and Galois
Abel proved the quintic formula impossible, then turned to elliptic functions and died of tuberculosis at twenty-six. Galois pushed both ideas further. His submitted papers were rejected, and what he wrote in prison was not a complete proof. It argued for another layer of abstraction: instead of studying the formulas themselves, study the symmetries underneath them. He died in a duel after trying to publish five times, having asked his brother and a close friend to get his notes to Gauss and the other important mathematicians of the day.
A century to be believed
The notes sat for twenty years until Liouville saw something in them and tried to clean them up. Another twenty years or so passed before Jordan put together a treatment close to modern group theory that was attributed to Galois. Real uses took longer still. In the twentieth century, Gell-Mann anticipated quarks from a purely group-theoretic question about how certain groups break down.
Why can't the quintic have a formula?
The roots of a general degree-five equation can be shuffled among themselves in 120 ways, and those shuffles form a group called S5. A formula built from roots and arithmetic corresponds to peeling that group down in simple stages, and S5 contains a piece called A5 that cannot be broken down further. A5 has 60 elements and is the smallest non-abelian simple group. The degree-two, three, and four groups all peel apart cleanly, which is why they have formulas.
How did group theory help predict quarks?
In the early 1960s, Murray Gell-Mann sorted the known particles called hadrons into families using the symmetry group SU(3). The pattern had a gap, and it called for a particle with a specific charge and strangeness that nobody had seen. That particle, the omega-minus, was found in a bubble chamber at Brookhaven National Laboratory in 1964. The same SU(3) pattern led Gell-Mann and George Zweig to propose quarks as its building blocks.
How did Galois's notes finally reach the public?
Joseph Liouville presented Galois's manuscripts to the French Academy in 1843 and published them in his Journal de mathematiques pures et appliquees in 1846. Camille Jordan then gave the theory its first full treatment in his 1870 book Traite des substitutions. That book set out permutation groups and their structure in the form later mathematicians built on.
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