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Counting Schur Rings over Cyclic Groups of Semi-prime Order

This paper extends the enumeration of Schur rings over cyclic groups by providing a count for those of semi-prime order $pq$ (where $p$ and $q$ are distinct primes) and order $4p$ .

Original authors: Joseph Keller, Andrew Misseldine, Max Sullivan

Published 2026-06-17

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Joseph Keller, Andrew Misseldine, Max Sullivan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a circular table with a specific number of seats, say $n$ seats. In the world of mathematics, this is called a cyclic group. Now, imagine you want to organize the people sitting at this table into different "clubs" or "teams" based on how they relate to one another.

This paper is about counting exactly how many different ways you can organize these teams, following a very strict set of rules. The authors call these organized structures Schur Rings.

Here is the breakdown of what the paper does, using simple analogies:

The Rules of the Game

To count these "team arrangements" (Schur Rings), the paper relies on a few basic rules:

The Leader: One team must always be just the person sitting in seat #1 (the identity).
The Mirror: If a team includes someone, it must also include their "mirror image" (their mathematical inverse).
The Mix-and-Match: If you take two teams and mix their members together in all possible ways, the result must be a new combination of existing teams.

The Main Challenge: Two Types of Tables

The authors focus on two specific types of tables (groups) that are built from prime numbers (numbers divisible only by 1 and themselves):

The "Semiprime" Table ($pq$): A table with $p \times q$ seats, where $p$ and $q$ are two different prime numbers.
The "Four Times a Prime" Table ( $4p$ ): A table with $4 \times p$ seats.

The goal was to write a "recipe" (a formula) that tells you exactly how many valid team arrangements exist for any table of these sizes.

The Four Building Blocks

The paper explains that every possible team arrangement is built from one of four "families" of structures. Think of these like different ways to build a house:

The Trivial House: The simplest arrangement. Everyone is either in the "Leader's Club" or in the "Everyone Else" club. There is only one way to do this.
The Direct Product House: Imagine two smaller tables side-by-side. You can arrange the teams on the left table and the teams on the right table independently, then combine them.
The Wedge House: This is a bit more complex. It's like taking a smaller arrangement and "gluing" it onto a larger one in a specific way. The paper has to be very careful here to make sure they don't count the same house twice just because it was glued together in a different order.
The Automorphic House: This is the most mathematical part. It relies on the "symmetry" of the table. If you can rotate or flip the table in certain ways without changing the pattern, those symmetries create new team arrangements. The paper notes that counting these is the same as counting the number of "sub-groups" (smaller symmetry groups) inside the table's symmetry group.

The Big Discovery: The Formulas

The authors spent the paper deriving mathematical formulas to count these arrangements for the two table types mentioned above.

For the $pq$ table: They found a formula that looks at the "prime ingredients" of $p-1$ and $q-1$ . It's like saying, "To know how many ways you can arrange a table of 21 seats ( $3 \times 7$ ), you need to look at the factors of 2 ($3-1$) and 6 ($7-1$)."
- Example: For a table of 21 seats, there are exactly 27 different valid team arrangements.
For the $4p$ table: They found a similar formula, but it's slightly more complicated because the number 4 adds extra layers of symmetry.
- Example: For a table of 12 seats ( $4 \times 3$ ), there are exactly 32 different valid team arrangements.

Why This Matters (According to the Paper)

The paper mentions that these "team arrangements" (Schur Rings) are connected to algebraic graph theory and association schemes. In plain English, this means they are used to understand how points in a network (like a social network or a computer network) are connected. By counting these rings, mathematicians are essentially counting the possible "shapes" or "patterns" that these networks can take.

Summary

The paper is a counting exercise. It takes two specific, somewhat complex types of mathematical circles (groups of size $pq$ and $4p$ ) and provides a precise calculator (formula) to tell you exactly how many valid ways you can organize their internal structures. It does this by breaking every possible organization down into four basic building blocks and carefully making sure no duplicates are counted.

The authors also verified their math by running computer programs to check all tables up to a certain size, and their formulas matched the computer's count perfectly.

Technical Summary: Counting Schur Rings over Cyclic Groups of Semiprime Order

Problem Statement
This paper addresses the enumeration problem of Schur rings over finite cyclic groups, specifically focusing on groups of semiprime order $n = pq$ (where $p$ and $q$ are distinct primes) and groups of order $n = 4p$ (where $p$ is an odd prime). Let $\Omega(n)$ denote the number of distinct Schur rings over the cyclic group $Z_n$ . While previous work by the second author established general techniques for enumerating cyclic Schur rings, this paper provides explicit closed-form formulas for $\Omega(pq)$ and $\Omega(4p)$ , extending the known enumeration up to order 400.

Methodology
The authors employ the classification framework established by Leung and Man, known as the Fundamental Theorem of Schur Rings over Cyclic Groups. This theorem states that all Schur rings over cyclic groups belong to one of four families:

Trivial Schur rings: The ring generated by $\{1, G \setminus \{1\}\}$ .
Automorphic Schur rings: Rings associated with subgroups of the automorphism group $\text{Aut}(Z_n)$ .
Direct products: Rings formed by the tensor product of Schur rings over direct factors of the group.
Wedge products: Rings constructed via a "wedge" operation over a proper section $U = [K, H]$ of the group, requiring specific compatibility conditions.

The methodology involves a recursive counting strategy based on the "wedge core" of a Schur ring. A Schur ring is decomposable if it can be expressed as a wedge product; otherwise, it is indecomposable. The total count $\Omega(n)$ is derived by summing the counts of:

The single trivial ring.
Direct products (which, in the semiprime case, are often automorphic).
Wedge-decomposable rings (counted by analyzing proper sections and compatible subrings).
Indecomposable automorphic rings (counted by determining the number of subgroups of $\text{Aut}(Z_n)$ ).

To count the automorphic rings, the paper utilizes the isomorphism $\text{Aut}(Z_n) \cong \prod \text{Aut}(Z_{p_i^{e_i}})$ . The authors apply a lemma by C˘alug˘areanu regarding the number of subgroups of a direct product of cyclic $p$ -groups, $Z_{p^k} \times Z_{p^\ell}$ , to compute $|L(\text{Aut}(Z_n))|$ .

Key Contributions and Results

1. The Semiprime Case ($n = pq$)
The paper derives a general formula for $\Omega(pq)$ based on the prime factorizations of $p-1$ and $q-1$ .

Theorem 1.1: Let $p = \prod_{i=1}^n r_i^{k_i} + 1$ and $q = \prod_{i=1}^n r_i^{\ell_i} + 1$ , where $\{r_i\}$ are distinct primes. The number of Schur rings is given by:
$\Omega(pq) = \left( \prod_{i=1}^n \sum_{j=0}^{\min(k_i, \ell_i)} \phi(r_i^j)(k_i - j + 1)(\ell_i - j + 1) \right) + 2 \prod_{i=1}^n (k_i + 1)(\ell_i + 1) + 1$
Here, the first term counts the automorphic rings (via the subgroup lattice of the automorphism group), the second term counts the wedge-decomposable rings, and the $+1$ accounts for the trivial ring.
Corollaries: The paper provides simplified formulas for specific cases, including:
- Safe Primes: If $p$ and $q$ are safe primes ( $p=2r+1$ ), $\Omega(pq) = 53$ .
- Fermat Primes: Specific simplifications are provided for $p=3$ and $p=5$ .
- General $2p$ : $\Omega(2p) = 3x + 1$ , where $x$ is the number of divisors of $p-1$ .

2. The Order $4p$ Case
The paper addresses the more complex case where the group order is $4p$ .

Theorem 1.6: Let $p = 2^k a + 1$ (with $a$ odd) and $x$ be the number of divisors of $p-1$ . The number of Schur rings is:
$\Omega(4p) = \frac{15k + 14}{k + 1}x + 3$
The proof involves a detailed analysis of wedge cores of orders $2, p, 4,$ and $2p$ , carefully accounting for overlaps between wedge products and direct products to avoid double-counting.
Special Cases: For Fermat primes ( $p=2^k+1$ ), the formula simplifies to $\Omega(4p) = 15k + 17$ . For safe primes, $\Omega(4p) = 61$ .

Significance and Claims
The paper positions itself as a continuation of the work initiated by the second author in previous publications [4, 5, 6]. Its primary significance lies in providing exact, computable formulas for $\Omega(n)$ in the semiprime and $4p$ cases, which were previously only accessible via computer enumeration for small orders.

Validation: The authors note that their formulas align perfectly with previous computer-generated enumerations of Schur rings over groups up to order 63 (by Ziv-Av [8]) and up to order 400 (by the second author [4]).
Scope: The paper does not claim to solve the enumeration problem for all cyclic groups but restricts its contribution to the specific structural cases of $pq$ and $4p$ . It provides a comprehensive table of values for all such semiprimes under 100, demonstrating the utility of the derived formulas.
Theoretical Context: The work reinforces the connection between Schur rings, association schemes, and algebraic graph theory, utilizing the structural properties of the automorphism group to reduce the combinatorial complexity of the enumeration.