Datasets:
TC⁰ Groups (Solvable)
This directory contains all solvable permutation groups from our dataset. These groups can theoretically be solved by constant-depth threshold circuits (TC⁰), which means models like Transformers and standard SSMs should be able to learn them.
Groups in this category:
Small Symmetric/Alternating (Solvable)
- S3 (order 6)
- S4 (order 24)
- A3 (order 3)
- A4 (order 12)
Cyclic Groups (All solvable abelian)
- C3 through C30
Dihedral Groups (All solvable)
- D3 through D20
Other Solvable Groups
- V4: Klein four-group (order 4)
- Q8, Q16, Q32: Quaternion groups
- Elementary abelian groups (Z_p^k)
- F20, F21: Frobenius groups
Key Property
All these groups have a composition series with abelian quotients, making them solvable and placing them in TC⁰.