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https://github.com/bolner/finite-group-explorer
Heuristic search algorithms for abelian and non-abelian small groups for educational purposes.
https://github.com/bolner/finite-group-explorer
cayley cpp graph graphviz group-theory
Last synced: 19 days ago
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Heuristic search algorithms for abelian and non-abelian small groups for educational purposes.
- Host: GitHub
- URL: https://github.com/bolner/finite-group-explorer
- Owner: bolner
- License: apache-2.0
- Created: 2020-06-07T13:17:04.000Z (over 4 years ago)
- Default Branch: master
- Last Pushed: 2020-06-30T14:01:51.000Z (over 4 years ago)
- Last Synced: 2024-10-28T10:44:58.161Z (2 months ago)
- Topics: cayley, cpp, graph, graphviz, group-theory
- Language: C++
- Size: 60.5 KB
- Stars: 0
- Watchers: 2
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
Awesome Lists containing this project
README
Finite Group Explorer
=====================
Heuristic search algorithms for [abelian](https://en.wikipedia.org/wiki/Abelian_group) and [non-abelian](https://en.wikipedia.org/wiki/Non-abelian_group) [small groups](https://en.wikipedia.org/wiki/List_of_small_groups).
My main motivation was the lack of similar heuristic programs on the internet. I was curious which are the biggest
groups that can be found this way, and how soon would one reach a combinatorial explosion on current CPUs. (This limit seems to be around order 10 for random search and 16-20 for the systematic.)
# Modules
The underlying representation of the groups in each module is done by using [Cayley tables](https://en.wikipedia.org/wiki/Cayley_table). In addition to the Cayley table, the search algorithms are making use of bitmasks that summarize values appearing on a given row or column (etc.). This technique is used frequently in Chess engines.
| Module | Description |
| --- | --- |
| [LatinHeuristics.hpp](./LatinHeuristics.hpp) | Searches for [reduced latin squares](https://en.wikipedia.org/wiki/Latin_square#Reduced_form) and disregards the [associative rule](https://en.wikipedia.org/wiki/Group_(mathematics)#Definition). Its findings might be either quasigroups or groups when associativity appears by chance. |
| [AssocHeuristics.hpp](./AssocHeuristics.hpp) | Searches for proper groups by using the associative rule too. The results can be both abelian and non-abelian. |
| [RandomHeuristics.hpp](./RandomHeuristics.hpp) | Same as AssocHeuristics but the search is randomized. This has much worse performance. |
| [CycleGraph.hpp](./CycleGraph.hpp) | Can generate the [Graphviz](https://dreampuf.github.io/GraphvizOnline/) and the [CsAcademy](https://csacademy.com/app/graph_editor/) code of the [Cycle Graph](https://en.wikipedia.org/wiki/Cycle_graph_(algebra)) of a group. Can also list the cyclic subgroups of the group. |
| [Classifier.hpp](./Classifier.hpp) | Checks for properties of the group. Now supports: Associative, Abelian, Cyclic, Simple, Dedekind, Hamiltonian. Can list the subgroups and normal subgroups. |
# 1. Example result: A4
A4 non-abelian, alternating group, order 12.
## 1.1. Cycle graph
## 1.2. Cayley table
| * |1|2|3|4|5|6|7|8|9|10|11|12|
| - | - | - | - | - | - | - | - | - | - | - | - | - |
|1|1|2|3|4|5|6|7|8|9|10|11|12|
|2|2|1|4|3|6|5|8|7|10|9|12|11|
|3|3|4|1|2|7|8|5|6|11|12|9|10|
|4|4|3|2|1|8|7|6|5|12|11|10|9|
|5|5|7|8|6|9|11|12|10|1|3|4|2|
|6|6|8|7|5|10|12|11|9|2|4|3|1|
|7|7|5|6|8|11|9|10|12|3|1|2|4|
|8|8|6|5|7|12|10|9|11|4|2|1|3|
|9|9|12|10|11|1|4|2|3|5|8|6|7|
|10|10|11|9|12|2|3|1|4|6|7|5|8|
|11|11|10|12|9|3|2|4|1|7|6|8|5|
|12|12|9|11|10|4|1|3|2|8|5|7|6|
# 2. Example result: G166
Abelian, order 16, product. Z4 x Z22
## 2.1. Cycle graph
## 2.2. Generated Graphviz code of the graph
```dot
strict graph Group {
node [shape=circle, fontsize=6, fixedsize=true, width=0.2]
1 [style=filled]
1 -- 9 -- 3 -- 11 -- 1
1 -- 10 -- 3 -- 12 -- 1
1 -- 13 -- 3 -- 15 -- 1
1 -- 14 -- 3 -- 16 -- 1
1 -- 2 -- 1
1 -- 4 -- 1
1 -- 5 -- 1
1 -- 6 -- 1
1 -- 7 -- 1
1 -- 8 -- 1
}
```
## 2.3. Cayley table
| | | | | | | | | | | | | | | | |
| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
|1|2|3|4|5|6|7|8|9|10|11|12|13|14|15|16|
|2|1|4|3|6|5|8|7|10|9|12|11|14|13|16|15|
|3|4|1|2|7|8|5|6|11|12|9|10|15|16|13|14|
|4|3|2|1|8|7|6|5|12|11|10|9|16|15|14|13|
|5|6|7|8|1|2|3|4|13|14|15|16|9|10|11|12|
|6|5|8|7|2|1|4|3|14|13|16|15|10|9|12|11|
|7|8|5|6|3|4|1|2|15|16|13|14|11|12|9|10|
|8|7|6|5|4|3|2|1|16|15|14|13|12|11|10|9|
|9|10|11|12|13|14|15|16|3|4|1|2|7|8|5|6|
|10|9|12|11|14|13|16|15|4|3|2|1|8|7|6|5|
|11|12|9|10|15|16|13|14|1|2|3|4|5|6|7|8|
|12|11|10|9|16|15|14|13|2|1|4|3|6|5|8|7|
|13|14|15|16|9|10|11|12|7|8|5|6|3|4|1|2|
|14|13|16|15|10|9|12|11|8|7|6|5|4|3|2|1|
|15|16|13|14|11|12|9|10|5|6|7|8|1|2|3|4|
|16|15|14|13|12|11|10|9|6|5|8|7|2|1|4|3|
# 3. Example result: Dih4
Dih4: non-abelian, dihedral group, order 8, extraspecial group, nilpotent.
## 3.1. Cycle graph
## 3.2. Generated Graphviz code of the graph
```dot
strict graph Group {
node [shape=circle, fontsize=6, fixedsize=true, width=0.2]
1 [style=filled]
1 -- 2 -- 1
1 -- 3 -- 6 -- 7 -- 1
1 -- 4 -- 1
1 -- 5 -- 1
1 -- 7 -- 6 -- 3 -- 1
1 -- 8 -- 1
}
```
## 3.3. Cayley table
| | | | | | | | |
| - | - | - | - | - | - | - | - |
|1|2|3|4|5|6|7|8|
|2|1|4|3|6|5|8|7|
|3|8|6|2|4|7|1|5|
|4|7|5|1|3|8|2|6|
|5|6|8|7|1|2|4|3|
|6|5|7|8|2|1|3|4|
|7|4|1|5|8|3|6|2|
|8|3|2|6|7|4|5|1|
# 4. Example result: G24
Abelian, order 24. This was found by accident, when I forgot to stop a program and it ran for around an hour.
## 4.1. Cycle graph
## 4.2. Generated Graphviz code of the graph
```dot
strict graph Group {
node [shape=circle, fontsize=6, fixedsize=true, width=0.2]
1 [style=filled]
1 -- 10 -- 17 -- 2 -- 9 -- 18 -- 1
1 -- 11 -- 17 -- 3 -- 9 -- 19 -- 1
1 -- 12 -- 17 -- 4 -- 9 -- 20 -- 1
1 -- 13 -- 17 -- 5 -- 9 -- 21 -- 1
1 -- 14 -- 17 -- 6 -- 9 -- 22 -- 1
1 -- 15 -- 17 -- 7 -- 9 -- 23 -- 1
1 -- 16 -- 17 -- 8 -- 9 -- 24 -- 1
}
```
## 4.3. Cayley table
| | | | | | | | | | | | | | | | | | | | | | | | |
| - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - | - |
|1|2|3|4|5|6|7|8|9|10|11|12|13|14|15|16|17|18|19|20|21|22|23|24|
|2|1|4|3|6|5|8|7|10|9|12|11|14|13|16|15|18|17|20|19|22|21|24|23|
|3|4|1|2|7|8|5|6|11|12|9|10|15|16|13|14|19|20|17|18|23|24|21|22|
|4|3|2|1|8|7|6|5|12|11|10|9|16|15|14|13|20|19|18|17|24|23|22|21|
|5|6|7|8|1|2|3|4|13|14|15|16|9|10|11|12|21|22|23|24|17|18|19|20|
|6|5|8|7|2|1|4|3|14|13|16|15|10|9|12|11|22|21|24|23|18|17|20|19|
|7|8|5|6|3|4|1|2|15|16|13|14|11|12|9|10|23|24|21|22|19|20|17|18|
|8|7|6|5|4|3|2|1|16|15|14|13|12|11|10|9|24|23|22|21|20|19|18|17|
|9|10|11|12|13|14|15|16|17|18|19|20|21|22|23|24|1|2|3|4|5|6|7|8|
|10|9|12|11|14|13|16|15|18|17|20|19|22|21|24|23|2|1|4|3|6|5|8|7|
|11|12|9|10|15|16|13|14|19|20|17|18|23|24|21|22|3|4|1|2|7|8|5|6|
|12|11|10|9|16|15|14|13|20|19|18|17|24|23|22|21|4|3|2|1|8|7|6|5|
|13|14|15|16|9|10|11|12|21|22|23|24|17|18|19|20|5|6|7|8|1|2|3|4|
|14|13|16|15|10|9|12|11|22|21|24|23|18|17|20|19|6|5|8|7|2|1|4|3|
|15|16|13|14|11|12|9|10|23|24|21|22|19|20|17|18|7|8|5|6|3|4|1|2|
|16|15|14|13|12|11|10|9|24|23|22|21|20|19|18|17|8|7|6|5|4|3|2|1|
|17|18|19|20|21|22|23|24|1|2|3|4|5|6|7|8|9|10|11|12|13|14|15|16|
|18|17|20|19|22|21|24|23|2|1|4|3|6|5|8|7|10|9|12|11|14|13|16|15|
|19|20|17|18|23|24|21|22|3|4|1|2|7|8|5|6|11|12|9|10|15|16|13|14|
|20|19|18|17|24|23|22|21|4|3|2|1|8|7|6|5|12|11|10|9|16|15|14|13|
|21|22|23|24|17|18|19|20|5|6|7|8|1|2|3|4|13|14|15|16|9|10|11|12|
|22|21|24|23|18|17|20|19|6|5|8|7|2|1|4|3|14|13|16|15|10|9|12|11|
|23|24|21|22|19|20|17|18|7|8|5|6|3|4|1|2|15|16|13|14|11|12|9|10|
|24|23|22|21|20|19|18|17|8|7|6|5|4|3|2|1|16|15|14|13|12|11|10|9|
# TODO
- Add more tests into the classifier.
- Find a decent graph rendering tool.