Number of groups of a given order¶
The function assigning each positive integer n the number of isomorphism classes of finite groups with exactly n elements.
Core Idea¶
For a fixed positive integer n, the problem asks how many abstract group structures of cardinality n exist, counting isomorphic realizations once. It is a classification count, not the order of one group or a count of its subgroups.
Factorization of n and structural theorems constrain the answer. Prime orders have one cyclic type and prime-square orders have two abelian types, while higher prime powers show rapid growth; exact counts require completeness as well as constructions.
Scope of Application¶
- Small-order classification. Enumerates and separates all types for a fixed n.
- Prime-power groups. Studies rapidly growing families at orders p to a power.
- Computational algebra. Constructs catalogs and tests isomorphism for feasible orders.
- Asymptotic enumeration. Bounds growth across families of orders.
- Structural group theory. Uses Sylow, solvability, extension, and action constraints.
Clarity¶
State n, whether the result is exact or asymptotic, the isomorphism convention, the admitted class of groups, the completeness argument, and how non-isomorphism is certified. Distinguish known bounds from exact enumeration. Inclusion test: Require a fixed positive integer n and a count of all pairwise non-isomorphic finite groups having exactly n elements, supported by a complete classification, enumeration, or theorem for that order. Exclusion test: Exclude the number of elements in one group, the number of subgroups of a group, counts of labeled Cayley tables, counts restricted to abelian or simple groups unless stated, and asymptotic estimates presented as exact values. Nearest boundary: Order of a group measures the elements in one group; the number of groups of order n counts distinct abstract group structures sharing that order. Exit condition: The identity fails when isomorphic relabelings are counted separately or when the collection is restricted without changing the stated function. Common misclassifications: It is not the order of a particular group. It is not a count of subgroups inside one group. It is not a count of labeled multiplication tables. It is not the classification of finite simple groups alone. Nearest named distinctions: Order of a Group: Order is the cardinality of one group; this abstraction counts group types sharing that cardinality. Number of Subgroups: Subgroup counts depend on one group's internal lattice. Finite Simple Group Classification: Simple groups are building blocks, but most groups counted at an order are not simple. Labeled Cayley Tables: Relabelings can encode the same isomorphism class and must be identified. Asymptotic Group Enumeration: Asymptotic growth estimates need not give the exact value at a specified n.
Manages Complexity¶
The counting function compresses a classification problem into an integer while retaining a strong equivalence convention. Its value is meaningful only when every finite group of order n is represented and duplicate presentations have been identified.
Abstract Reasoning¶
- Fix the positive integer n.
- Factor n and derive structural restrictions.
- Construct candidate groups through presentations, products, actions, or extensions.
- Verify the group axioms and order for each candidate.
- Quotient the collection by group isomorphism.
- Prove completeness and report the resulting count or qualified bound.
Knowledge Transfer¶
The transferable cargo is counting structures of fixed size modulo isomorphism. It transfers to rings, graphs, and other finite structures when their carrier, axioms, size, and equivalence relation are retyped; group-specific counts and theorems do not transfer.
Relationships to Other Abstractions¶
Current abstraction Number of groups of a given order Domain-specific
Parents (1) — more general patterns this builds on
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Number of groups of a given order is a kind of Function (Mapping) Prime
It is a counting function from positive integers to nonnegative integers.
Hierarchy path (1) — routes to 1 parentless root
- Number of groups of a given order → Function (Mapping)
Neighborhood in Abstraction Space¶
Number of groups of a given order sits in a crowded region of the domain-specific corpus (32nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Group Structure & Subgroup Properties (10 abstractions)
Nearest neighbors
- Order (group theory) — 0.92
- Group code — 0.88
- Cycle Graph (Algebra) — 0.88
- Free Group — 0.88
- Finiteness Properties of Groups — 0.88
Computed from structural-signature embeddings · 2026-10-08