A finite cabinet, with an infinite claim

The List With Nothing After It

One list says every finite atom of symmetry belongs to 18 infinite families or 26 sporadic exceptions. Test a small entry yourself, then inspect the repaired proof branch that makes the word every carry its weight.

The object

Open the drawers

The headings continue through parameters. The sporadic names do not. Search a notation, a family, or a name.

Preparing the catalog

Layer 1, a finite certificate

Try to keep one piece small

A normal subgroup must keep every conjugate of any element it contains. Choose a nonidentity even permutation. The machine adds all conjugates, then closes under multiplication and inverses.

Choose an element, then take its normal closure.

The identity is absent on purpose. Its normal closure is the trivial subgroup by definition, so it is no evidence of simplicity.

Sanity check loading. In S4 the same algorithm must be able to stop at a proper normal subgroup.

Each light is one even permutation of five letters. The layout is an index, not a Cayley diagram. The lit set itself is computed from the selected seed.

Layer 2, split the same group three ways

Walk down S4

At each step, choose a maximal proper normal subgroup. The quotient is built as a coset multiplication table and tested for simplicity before the route may continue.

Computing the available normal steps.

The route begins at S4.

A composition factor list does not reconstruct the original group. Different extensions can have the same factors.

No quotient chosen yet

Choose the next normal subgroup to construct a quotient table.

The word every lives elsewhere

Pull the quasithin volume

This ledger is sourced reading aid. It is not a formal proof graph. Remove one cited branch and watch the claim lose support at that branch.

Literature ledger connected at the quasithin branch.

Catalog statement

Every finite simple group belongs to the listed types. This global claim is cited. Nothing in the browser recomputes it.

Boya 2011 for the accounting. Aschbacher 2004 for theorem status.

Quasithin branch

Two 2004 volumes. Reading official catalog lengths.

AMS Mathematical Surveys and Monographs 111 and 112.

Revision status

Reading the official numbered series fixture.

The second-generation series has an aim, not a completed-volume claim.

The check

The browser runs these finite computations from permutation definitions. The offline verifier starts again with a different permutation generator, checks the group laws, and compares the results and checksums.

Catalog audit loading

Structured records, de-duplicated live.

A5 audit loading

Every nonidentity normal closure, not only your selected seed.

S4 audit loading

All subgroups, all composition routes, every quotient table.

Checksums loading

FNV-1a transcription checksums, not cryptographic proofs.

Boundary. The A5 calculation proves that this displayed group is simple. The S4 calculation makes Jordan-Holder operable for this displayed group. The catalog count proves only that this page contains the promised records. Completeness of the classification remains a literature-backed theorem, not a browser result and not a proof-assistant formalization.

Uncertainties and free choices
  • Catalog convention. The free grouping choice is one cyclic family, one alternating family, and sixteen Lie-type families. The Tits group is placed on the Lie-type side. Other presentations can group headings differently.
  • Names and aliases. Family labels are compact headings, not parameter restrictions. J2 is also called Hall-Janko. The compact display uses Fi24' for the simple Fischer group. These display choices do not change the count.
  • Finite model choices. Permutations compose right to left. Coset representatives are chosen by enumeration order. The chosen A5 seed and the chosen order-two subgroup in V4 are yours. None changes the final certificate.
  • Vacuous settings. The identity seed is disabled because its trivial closure is automatic. Merely counting catalog rows and toggling the ledger are also tautological. The proper closure in S4 shows the normal-closure algorithm can stop short.
  • Proof ledger. Its arrows are an editorial map of cited dependency, not machine verification. Removing a node changes whether this page displays support. It does not change the theorem.
  • Historical scope. About 100 contributors and estimates from approximately 10,000 to 15,000 pages use different scopes. They are estimates, not a census. This page does not use them as computed quantities.
  • Repair wording. The two quasithin volumes appeared in 2004. Aschbacher's contemporary status report used cautious language. A treatment of an omitted standard-component case appeared in 2008, so this page does not call 2004 the final repair without qualification.
  • Revision status. The official catalog check on 2026-07-29 found ten numbered GLS volumes, the latest dated 2023. Failure to find a later numbered volume is a dated catalog result, not proof that no unpublished work exists.

Offline command: node research/finite-simple-groups/verify-finite-simple-groups.mjs

Sources held against the object

  1. Luis J. Boya, Introduction to Sporadic Groups, SIGMA 7:009 (2011), DOI 10.3842/SIGMA.2011.009. The arXiv page was checked field by field: title, sole author, submitted 16 January 2011, and one version. Its abstract supplies both catalog equations.
  2. Michael Aschbacher, The Status of the Classification of the Finite Simple Groups, Notices of the AMS 51(7), 2004. This supplies the Jordan-Holder statement, theorem-status caution, and the description of the quasithin work as roughly 1,200 pages.
  3. Michael Aschbacher and Stephen D. Smith, The Classification of Quasithin Groups I and II, AMS, 2004. The official catalog gives 477 and 743 pages and describes a two-part book.
  4. Koichiro Harada and Ronald Solomon, Finite groups having a standard component L of type M12-hat or M22-hat, Journal of Algebra 319(2), 621-628, 2008.
  5. Inna Capdeboscq, Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The Classification of the Finite Simple Groups, Number 10, AMS, 2023. The publisher calls it the tenth in a series whose aim is to provide a complete proof.
  6. Daniel Gorenstein, The Classification of Finite Simple Groups, Volume 1, Springer, 1983, and Gorenstein, Lyons, and Solomon, The Classification of the Finite Simple Groups, Number 7, AMS, 2018. Their publisher descriptions supply the differently scoped estimates of about 100 contributors and approximately 10,000 to an estimated 15,000 pages.