Group Ring Subalgebras and Möbius Counting

ISEF Category: Mathematics

Subcategory: Algebra  ·  Difficulty: Advanced  ·  Setup: University Lab  ·  Time: Full Year

The Hook

A tiny change in a group can make its ring behave like a whole new algebraic world. That means a structure with just a few dozen elements can still hide a huge web of subalgebras. You can map that web, count it, and hunt for a pattern that works across whole families. That is the kind of project that turns abstract algebra into real research.

What Is It?

A group ring mixes two ideas, a group and a ring. Think of a group as a set of symmetries, and a ring as a system where you can add and multiply. When you build the group ring F_p[G], you are combining the symmetry data from G with coefficients from the finite field F_p. The result is a new algebraic object with its own subalgebras, which are smaller pieces closed under addition and multiplication.

Your job is to study how those subalgebras fit together. The whole collection forms a lattice, which means you can organize the subalgebras by inclusion, like folders inside folders. A Möbius function on that lattice is a counting tool that often reveals hidden structure. If you can predict the number of subalgebras from that Möbius data, then you are not just counting objects, you are finding a rule behind the count.

Why This Is a Good Topic

This is a strong science fair topic because it starts with a clear question, but still leaves room for original work. You can generate data with GAP or SageMath, look for patterns across non-abelian groups of order 24 or less, and test whether one formula fits many cases. The project connects pure math, computation, and proof. You can also split the work into levels, first data collection, then pattern spotting, then a proof for one family such as dihedral or dicyclic groups.

Research Questions

• How does the number of subalgebras in F_p[G] change as p changes for a fixed non-abelian group G of order ≤ 24?
• What is the effect of choosing a dihedral group versus a dicyclic group on the lattice size of F_p[G]?
• Does the Möbius function of the subalgebra lattice predict the total number of subalgebras for small group rings?
• To what extent do non-isomorphic groups of the same order produce different subalgebra lattices over the same field F_p?
• Which structural features of G, such as the number of cyclic subgroups, best explain the subalgebra count in F_p[G]?
• How does the subalgebra lattice of F_p[G] behave when G has more than one normal subgroup chain?
• To what extent can a formula proved for dihedral groups extend to other non-abelian families?

Basic Materials

• Computer with enough memory to run GAP or SageMath.
• GAP software installed.
• SageMath installed, or access to a SageMathCell or local SageMath setup.
• Spreadsheet software for tracking counts and patterns.
• Notebook for recording group labels, field choices, and subalgebra counts.
• Reference text on finite groups and basic ring theory.
• List of non-abelian groups of order ≤ 24 from GAP's SmallGroup library.

Advanced Materials

• University computer access or a strong personal laptop for repeated computations.
• GAP with the Subgroups and algebra packages you need for lattice exploration.
• SageMath with symbolic tools for testing conjectured formulas.
• LaTeX for writing proofs and documenting computations clearly.
• Optional Python scripts for batch runs, data cleanup, and plotting.
• Access to algebra textbooks or seminar notes on group rings and lattice theory.

Software & Tools

• GAP: Computes finite groups, subgroup data, and algebraic structures for small group rings.
• SageMath: Lets you combine symbolic algebra, scripting, and finite algebra experiments in one place.
• Python: Organizes output from GAP or SageMath and helps you compare counts across many cases.
• ImageJ: Not needed here, so skip it unless you decide to visualize printed lattice diagrams as images.
• LaTeX: Formats conjectures, tables, and proofs in a clean math-paper style.

Experiment Steps

1. Choose a narrow family of groups and one prime field, then state exactly what you will count.
2. Build a naming system for the groups, the fields, and the subalgebra data so your results stay organized.
3. Decide how you will represent each lattice, as a diagram, a table, or an adjacency-style record.
4. Test a few small examples by hand or with software, then check that your counting method matches the algebra.
5. Look for repeated patterns across related groups, and write a conjecture in a form that can be checked.
6. Plan a proof strategy for one family, then compare the proven case with the computer-generated data.

Common Pitfalls

• Mixing up subalgebras with subgroups, which makes the count meaningless for the ring you are studying.
• Skipping the field choice p, which can change the entire lattice and break your comparison across examples.
• Using different labels for the same group across software runs, which makes your dataset hard to trust.
• Counting duplicate structures twice after isomorphic relabeling, which inflates the subalgebra total.
• Trying to prove the full conjecture before you have a clean pattern, which usually leads to a messy and incomplete argument.

What Makes This Competitive

A strong version of this project does more than list examples. You would compare several non-isomorphic groups, test the same formula across many primes, and explain why the pattern works in one family but not another. Better yet, you would connect the computer output to a proof, not just a guess. Clear notation, careful case splitting, and a clean explanation of the Möbius argument would push the work much closer to research level.

Project Variations

• Study group rings over different prime fields and see whether the same lattice pattern survives.
• Compare dihedral, dicyclic, and quaternion-like groups to find which family gives the largest subalgebra count.
• Focus on a single invariant, such as the number of maximal subalgebras, and test whether it predicts the full lattice size.

Learn More

• GAP Manual: Search the official GAP documentation for finite group and subgroup commands, then use the SmallGroup library sections.
• SageMath Documentation: Search the SageMath docs for group rings, finite groups, and symbolic algebra examples.
• MIT OpenCourseWare, Algebra courses: Look for lecture notes on rings, fields, and groups to refresh the theory behind group rings.
• Journal of Algebra: Search for review articles and papers on group rings, lattice theory, and Möbius inversion in algebra.
• arXiv: Search for preprints on group rings, subgroup lattices, and algebraic combinatorics to see current methods.