Datasets:

BeeGass
/

Group-Theory-Collection

license: mit
task_categories:
  - text-generation
language:
  - en
tags:
  - mathematics
  - group-theory
  - permutations
  - symbolic-reasoning
  - algebra
  - sequence-modeling
pretty_name: Permutation Groups Composition Dataset
size_categories:
  - 10M<n<100M

Permutation Groups Composition Dataset

A comprehensive collection of permutation composition datasets for various mathematical groups including symmetric, alternating, cyclic, dihedral, and special groups, with multiple sequence length variants.

Dataset Description

This dataset contains permutation composition problems across 30 different mathematical groups with 8 different sequence length variants each, totaling 270 distinct configurations.

Supported Groups

Symmetric Groups (Sn)

S3 to S7: All permutations of n elements (orders: 6, 24, 120, 720, 5040)

Alternating Groups (An)

A3 to A7: Even permutations of n elements (orders: 3, 12, 60, 360, 2520)

Cyclic Groups (Cn/Zn)

C3 to C12: Cyclic groups of order n
Z3 to Z6: Alternative notation for cyclic groups
Orders: 3, 4, 5, 6, 7, 8, 10, 12

Dihedral Groups (Dn)

D3 to D8: Symmetries of regular n-gons (orders: 6, 8, 10, 12, 14, 16)

Special Groups

PSL(2,5): Projective special linear group (order 60)
F20: Frobenius group F(5,4) (order 20)

Length Variants

Each group is available with 8 different maximum sequence lengths:

2², 2³, 2⁴, 2⁵, 2⁶, 2⁷, 2⁸, 2⁹ (4, 8, 16, 32, 64, 128, 256, 512)

Each dataset consists of sequences of permutations that need to be composed to produce a target permutation. This is useful for:

Training models on algebraic reasoning and symbolic computation
Evaluating mathematical understanding and compositional generalization
Benchmarking sequence models on structured mathematical tasks
Studying group theory properties in neural networks
Research in abstract algebra and computational mathematics

Usage

from datasets import load_dataset

# NEW: Clean API - just specify group and max_len
s5_data = load_dataset(
    "BeeGass/permutation-groups",
    name="s5",          # Just the group name
    max_len=32,         # Optional: filter sequences ≤ 32 (default: 512)
    trust_remote_code=True
)

# More examples with the clean API
c8_short = load_dataset("BeeGass/permutation-groups", name="c8", max_len=16, trust_remote_code=True)
d4_medium = load_dataset("BeeGass/permutation-groups", name="d4", max_len=64, trust_remote_code=True)
all_short = load_dataset("BeeGass/permutation-groups", name="all", max_len=32, trust_remote_code=True)

# Backwards compatibility - old style still works
s5_data = load_dataset("BeeGass/permutation-groups", name="s5_data", trust_remote_code=True)
s5_len32 = load_dataset("BeeGass/permutation-groups", name="s5_len32", trust_remote_code=True)

# Access the data
train_data = s5_data["train"]
test_data = s5_data["test"]

# Example data point
print(train_data[0])
# {'input_sequence': '23 45 12', 'target': '67'}

Dataset Structure

Each example contains:

input_sequence: A space-separated sequence of permutation IDs to be composed
target: The ID of the resulting permutation after composition

The composition follows standard mathematical convention: for input [p1, p2, p3], the result is p3 ∘ p2 ∘ p1.

Available Configurations

Efficient Loading (Recommended)

Simply specify the group name and optional max_len parameter:

# Load any group with any max sequence length
dataset = load_dataset("BeeGass/permutation-groups", name="s5", max_len=32, trust_remote_code=True)

All Groups (30 total)

Group	Type	Order	Example Usage
S3-S7	Symmetric	6-5040	`name="s5", max_len=64`
A3-A7	Alternating	3-2520	`name="a4", max_len=32`
C3-C12	Cyclic	3-12	`name="c8", max_len=16`
Z3-Z6	Cyclic (alt)	3-6	`name="z5", max_len=128`
D3-D8	Dihedral	6-16	`name="d4", max_len=256`
PSL25	PSL(2,5)	60	`name="psl25", max_len=64`
F20	Frobenius	20	`name="f20", max_len=32`
all	Combined	-	`name="all", max_len=16`

Legacy Configuration Names

For backwards compatibility, old-style names still work:

s5_data (equivalent to name="s5")
s5_len32 (equivalent to name="s5", max_len=32)
etc.

Dataset Features

Variable sequence length: Input sequences range from 3 to maximum configured length
Length-specific variants: 8 different maximum lengths for each group (2² to 2⁹)
Consistent formatting: All permutations use space-separated integer IDs
Metadata included: Each dataset includes a metadata.json file mapping IDs to permutation array forms
Train/test split: 80/20 split for all configurations
Scaled sample sizes: Shorter sequences have more samples for efficient training

Understanding the Data

Each permutation is represented by a unique integer ID. The metadata.json file in each dataset folder provides the mapping from IDs to permutation array forms.

For example, in S3:

ID 0 might map to [0, 1, 2] (identity)
ID 1 might map to [0, 2, 1] (transpose elements 1 and 2)
etc.

Citation

If you use this dataset in your research, please cite:

@software{permutation_groups_dataset,
  author = {Bryan Gass},
  title = {Permutation Groups Dataset},
  year = {2024},
  publisher = {Hugging Face},
  url = {https://huggingface.co/datasets/BeeGass/permutation-groups}
}

Acknowledgments

This dataset was inspired by the work of William Merrill and his paper "The Illusion of State in State-Space Models", which explores the computational properties of state-space models through group theory.

License

This dataset is released under the MIT License.

Contact

For questions or issues, please open an issue on the GitHub repository.