libsemigroups · james-d-mitchell · Jun 2, 2026 · Jun 1, 2026 · Jun 1, 2026 · Jun 1, 2026
diff --git a/docs/case-studies/index.md b/docs/case-studies/index.md
diff --git a/docs/examples/atlas/index.md b/docs/examples/atlas/index.md
@@ -0,0 +1,185 @@
+# Verifying presentations in the ATLAS of Finite Groups
+
+In this example, we will verify the validity of the presentations of the
+sporadic groups as given in [this ATLAS][atlas].
+In particular, where possible, we will use the Todd-Coxeter algorithm to check
+that the size of group defined by the presentations is equal to the claimed
+size. Where the claimed size of the group is very large, we will instead
+calculate the size of small-index subgroups.
+
+??? info "libsemigroups_pybind11 version"
+
+    All examples provided on the subsequent subpages were run using
+    libsemigroups_pybind11 version 1.4.4 on a laptop with a 13th Gen Intel(R)
+    Core(TM) i7-13700H processor and 64 GB RAM.
+
+??? info "Defining groups with monoid presentations"
+
+    The presentations provided for the groups in the ATLAS are
+    *group presentations*. This means that it is assumed that there is
+    multiplicative identity $1$, and that each generator $a$ has an inverse
+    $a^{-1}$ such that $aa^{-1} = a^{-1}a = 1$. In [libsemigroups_pybind11][],
+    however, presentations are either *semigroup presentations* or
+    *monoid presentations*, depending whether the relations of the presentation
+    are allowed to contain the empty word $\varepsilon$. Therefore, we will need
+    to add extra generators and relations to a monoid presentation to define a
+    group.
+
+    Suppose that a group $G$ is defined by the group presentation
+    $\langle{A \mid R }\rangle_{\text{grp}}$. Then $G$ can also be defined by
+    the monoid presentation
+    $\langle{A \sqcup A^{-1} \mid R \cup R' }\rangle_{\text{mon}}$ where
+    $A^{-1}$ is a set disjoint from $A$ containing letters that will be treated
+    as inverses for the letters in $A$, and $R'$ is the set of relations of the
+    form $aa^{-1} = \varepsilon$ and $a^{-1}a = \varepsilon$.
+
+    In the subsequent subpages, we will use lowercase letters for the
+    generators that are given in the presentations in the ATLAS, and their
+    uppercase counterparts to represent their inverses. Therefore, many of our
+    examples will in a similar way to:
+
+    ```py
+    from libsemigroups_pybind11 import presentation, Presentation
+    p = presentation("abAB")
+    p.contains_empty_word(True)
+    presentation.add_inverse_rules(p, "ABab")
+    ```
+
+    The algorithms in [libsemigroups_pybind11][] were written for semigroups and
+    monoids. This means that there are no group-specific optimisations.
+
+The following tables summarise the results of this project. Click on a group
+to see more information.
+
+<div class="row">
+  <div class="column">
+    <div class="atlas-summary">
+      <table>
+        <tr>
+          <th colspan="5">Mathieu groups</th>
+        </tr>
+        <tr>
+          <td class="correct-size">
+            <a href="mathieu/m11">M<sub>11</sub></a>
+          </td>
+          <td class="correct-size">
+            <a href="mathieu/m12">M<sub>12</sub></a>
+          </td>
+          <td class="correct-size">
+            <a href="mathieu/m22">M<sub>22</sub></a>
+          </td>
+          <td class="correct-index">
+            <a href="mathieu/m23">M<sub>23</sub></a>
+          </td>
+          <td class="correct-index">
+            <a href="mathieu/m24">M<sub>24</sub></a>
+          </td>
+        </tr>
+      </table>
+    </div>
+    <div class="atlas-summary">
+      <table>
+        <tr>
+          <th colspan="7">Leech lattice groups</th>
+        </tr>
+        <tr>
+          <td class="could-not-verify"><a href="leech-lattice/hs">HS</a></td>
+          <td>
+            <a href="leech-lattice/j2">J<sub>2</sub></a>
+          </td>
+          <td class="no-presentation">Co<sub>1</sub></td>
+          <td>
+            <a href="leech-lattice/co2">Co<sub>2</sub></a>
+          </td>
+          <td class="no-presentation">Co<sub>3</sub></td>
+          <td><a href="leech-lattice/mcl">McL</a></td>
+          <td class="no-presentation">Suz</td>
+        </tr>
+      </table>
+    </div>
+    <div class="atlas-summary">
+      <table>
+        <tr>
+          <th colspan="8">Monster sections</th>
+        </tr>
+        <tr>
+          <td><a href="monster-sections/he">He</a></td>
+          <td class="no-presentation">HN</td>
+          <td class="no-presentation">Th</td>
+          <td>
+            <a href="monster-sections/fi22">Fi<sub>22</sub></a>
+          </td>
+          <td class="no-presentation">Fi<sub>23</sub></td>
+          <td class="no-presentation">Fi<sub>24</sub>'</td>
+          <td class="no-presentation">B</td>
+          <td class="no-presentation">M</td>
+        </tr>
+      </table>
+    </div>
+    <div class="atlas-summary">
+      <table>
+        <tr>
+          <th colspan="6">Pariahs</th>
+        </tr>
+        <tr>
+          <td>
+            <a href="pariahs/j1">J<sub>1</sub></a>
+          </td>
+          <td class="no-presentation">O'N</td>
+          <td>
+            <a href="pariahs/j3">J<sub>3</sub></a>
+          </td>
+          <td><a href="pariahs/ru">Ru</a></td>
+          <td>
+            <a href="pariahs/j4">J<sub>4</sub></a>
+          </td>
+          <td class="no-presentation">Ly</td>
+        </tr>
+      </table>
+    </div>
+    <div class="atlas-summary">
+      <table>
+        <tr>
+          <th>Miscellaneous</th>
+        </tr>
+        <tr>
+          <td><a href="misc/t">T</a></td>
+        </tr>
+      </table>
+    </div>
+  </div>
+
+  <div class="column">
+    <div class="atlas-summary">
+      <table>
+        <tr>
+          <th>Legend</th>
+        </tr>
+        <tr>
+          <td class="correct-size">
+            The presentation defines a group of the correct size
+          </td>
+        </tr>
+        <tr>
+          <td class="correct-index">
+            The presentation has a subgroup of the correct size
+          </td>
+        </tr>
+        <tr>
+          <td class="incorrect-size">
+            The presentation defines a group of the incorrect size
+          </td>
+        </tr>
+        <tr>
+          <td class="could-not-verify">Todd-Coxeter did not terminate</td>
+        </tr>
+        <tr>
+          <td class="no-presentation">No presentation is provided</td>
+        </tr>
+      </table>
+    </div>
+  </div>
+</div>
+
+[atlas]: https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/
+[libsemigroups_pybind11]: https://libsemigroups.github.io/libsemigroups_pybind11/index.html
diff --git a/docs/examples/atlas/leech-lattice/co2/index.md b/docs/examples/atlas/leech-lattice/co2/index.md
@@ -0,0 +1,12 @@
+# Conway group Co~2~
+
+> Order: $42,305,421,312,000$
+
+> Presentation: TODO
+
+On this page, we provide links to verifications that the maximal subgroups of
+the Conway group Co~2~ define groups of the correct order.
+
+## Maximal subgroups
+
+The following are maximal subgroups that have been verified:
diff --git a/docs/examples/atlas/leech-lattice/hs.md b/docs/examples/atlas/leech-lattice/hs.md
@@ -0,0 +1,28 @@
+# Higman-Sims group HS
+
+> Order: $44,352,000$
+
+> Presentation: TODO
+
+On this page, we verify that the above claimed presentation of the Higman-Sims group HS
+defines a group of order $44,352,000$.
+
+## The code
+
+=== "Python"
+
+    Coming soon!
+
+=== "ACE"
+
+    Coming soon!
+
+## The output
+
+=== "Python"
+
+    Coming soon!
+
+=== "ACE"
+
+    Coming soon!
diff --git a/docs/examples/atlas/leech-lattice/j2.md b/docs/examples/atlas/leech-lattice/j2.md
@@ -0,0 +1,28 @@
+# Janko group J~2~
+
+> Order: $604,800$
+
+> Presentation: TODO
+
+On this page, we verify that the above claimed presentation of the Janko group J~2~
+defines a group of order $604,800$.
+
+## The code
+
+=== "Python"
+
+    Coming soon!
+
+=== "ACE"
+
+    Coming soon!
+
+## The output
+
+=== "Python"
+
+    Coming soon!
+
+=== "ACE"
+
+    Coming soon!
diff --git a/docs/examples/atlas/leech-lattice/mcl/index.md b/docs/examples/atlas/leech-lattice/mcl/index.md
@@ -0,0 +1,12 @@
+# McLaughlin group McL
+
+> Order: $898,128,000$
+
+> Presentation: TODO
+
+On this page, we provide links to verifications that the maximal subgroups of
+the McLaughlin group McL define groups of the correct order.
+
+## Maximal subgroups
+
+The following are maximal subgroups that have been verified:
diff --git a/docs/examples/atlas/mathieu/m11.md b/docs/examples/atlas/mathieu/m11.md
@@ -0,0 +1,105 @@
+# Mathieu group M~11~
+
+> Order: $7,920$
+
+> Presentation:
+> $\langle{ a, b \mid a^2 = b^4 = (ab)^{11} = (ab^2)^6 = ababab^{−1}abab^2ab^{−1}abab^{−1}ab^{−1} = 1 }\rangle$
+
+On this page, we verify that the above claimed presentation of the Mathieu group
+M~11~ defines a group of order $7,920$.
+
+## The code
+
+=== "Python"
+
+    In [libsemigroups_pybind11][], the following script constructs the
+    presentation for M~11~ and runs the Todd-Coxeter algorithm.
+
+    ```python
+    from libsemigroups_pybind11 import Presentation, ToddCoxeter, congruence_kind, presentation
+    from libsemigroups_pybind11.words import parse_relations
+
+    # Setup the presentation object with the empty and inverses, so it can represent a group
+    p = Presentation("abAB")
+    p.contains_empty_word(True)
+    presentation.add_inverse_rules(p, "ABab")
+
+    # Add the defining relations
+    presentation.add_rule(p, parse_relations("a^2"), "")
+    presentation.add_rule(p, parse_relations("b^4"), "")
+    presentation.add_rule(p, parse_relations("(ab)^11"), "")
+    presentation.add_rule(p, parse_relations("(ab^2)^6"), "")
+    presentation.add_rule(p, parse_relations("ababaBabab^2aBabaBaB"), "")
+
+    # Run the Todd-Coxeter algorithm
+    tc = ToddCoxeter(congruence_kind.twosided, p)
+    tc.run()
+
+    assert tc.number_of_classes() == 7920
+    ```
+
+=== "ACE"
+
+    Coming soon!
+
+## The output
+
+=== "Python"
+
+    Running the above Python script produces the following output:
+
+    ```
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 START (strategy() = hlt)
+    #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 22], ∑(|u| + |v|) = 73
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: HLT 0.0 START
+    #0: ToddCoxeter: HLT 0.0.0        |       active |          killed |        defined
+    #0: ToddCoxeter: nodes            |            1 |               0 |              1
+    #0: ToddCoxeter:                  |       active |         missing |     % complete
+    #0: ToddCoxeter: edges            |            0 |               4 |           0.0%
+    #0: ToddCoxeter: time             | run 0 = 28µs | all runs = 28µs | elapsed = 92µs
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: HLT 0.0 STOP
+    #0: ToddCoxeter: HLT 0.0.1         |        active |           killed |         defined
+    #0: ToddCoxeter: nodes             |         7,920 |        1,992,427 |       2,000,347
+    #0: ToddCoxeter: diff 0.0.0        |        +7,919 |       +1,992,427 |      +2,000,346
+    #0: ToddCoxeter:                   |        active |          missing |      % complete
+    #0: ToddCoxeter: edges             |        31,680 |                0 |          100.0%
+    #0: ToddCoxeter: diff 0.0.0        |       +31,680 |               -4 |         +100.0%
+    #0: ToddCoxeter: phase 0.0 = 176ms | run 0 = 176ms | all runs = 176ms | elapsed = 176ms
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: HLT 0.1.2         |        active |           killed |         defined
+    #0: ToddCoxeter: nodes             |         7,920 |        1,992,427 |       2,000,347
+    #0: ToddCoxeter: diff 0.1.1        |            +0 |               +0 |              +0
+    #0: ToddCoxeter: diff 0.1.0        |        +7,919 |       +1,992,427 |      +2,000,346
+    #0: ToddCoxeter:                   |        active |          missing |      % complete
+    #0: ToddCoxeter: edges             |        31,680 |                0 |          100.0%
+    #0: ToddCoxeter: diff 0.1.1        |            +0 |               +0 |           +0.0%
+    #0: ToddCoxeter: diff 0.1.0        |       +31,680 |               -4 |         +100.0%
+    #0: ToddCoxeter: phase 0.1 = 176ms | run 0 = 176ms | all runs = 176ms | elapsed = 176ms
+    ++++++++++++++++++++++++++++++++
+    #0: ToddCoxeter: RUN 0 STOP (finished)
+    #0: ToddCoxeter: run 0                |     lookahead |       lookbehind |             hlt |       felsch
+    #0: ToddCoxeter: num. phases          |             0 |                0 |               1 |            0
+    #0: ToddCoxeter: time spent in phases |        - (0%) |           - (0%) |    176ms (100%) |       - (0%)
+    #0: ToddCoxeter: phase 0.1 = 176ms    | run 0 = 176ms | all runs = 176ms | elapsed = 176ms
+    ```
+
+=== "ACE"
+
+    Coming soon!
+
+:simple-ticktick: The computed size of the group matches the size of the group
+provided on [the ATLAS][atlas]: $7,920$.
+
+!!! note
+
+    The output you see when you run the script yourself might be different from
+    the above; the numbers produced and time taken will depend on the machine
+    you are using. However, the size of the group that is reported should be
+    the same as above.
+
+[atlas]: https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/
+[libsemigroups_pybind11]:
+  https://libsemigroups.github.io/libsemigroups_pybind11/index.html