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.
Structural Signature¶
Sig role-phrases:
- Positive integer n — Fixes the finite cardinality being classified. It is input. Counterfactual: Without a fixed n there is no value of the counting function.
- Finite group — Supplies a set of n elements with an associative operation, identity, and inverses. It is carrier. Counterfactual: Arbitrary magmas and semigroups are not counted.
- Group order — Tests that the underlying set has cardinality n. It is constraint. Counterfactual: Element order or subgroup order is a different quantity.
- Isomorphism relation — Collapses differently labeled multiplication tables representing the same abstract group. It is equivalence. Counterfactual: Counting presentations would overcount one group type.
- Structural restrictions — Use factorization and theorems to constrain candidate groups. It is inference. Counterfactual: Restrictions such as solvability do not by themselves enumerate all classes.
- Enumeration or proof — Establishes completeness and pairwise non-isomorphism of the counted list. It is validation. Counterfactual: A list of examples gives only a lower bound.
What It Is Not¶
- 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.
- Closest near-miss. 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.
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.
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.
Examples¶
Applied / In Practice¶
For prime p, every group of order p is cyclic, so the count of isomorphism classes is one.
Mapped back: input → p prime; count → 1.
Applied / In Practice¶
For p squared, the cyclic group and the product of two cyclic groups give exactly two isomorphism classes.
Mapped back: input → p^2; count → 2.
Applied / In Practice¶
Counting all subgroups of a fixed group of order n answers an internal incidence question, not the number of group isomorphism classes of order n.
Mapped back: object counted → subgroups.
Structural Tensions¶
T1 — Existence versus Complete Enumeration. Constructing several groups of order n does not show that none are missing.
Diagnostic: What completeness theorem closes the list?
T2 — Labeled Realizations versus Isomorphism Classes. Many multiplication tables differ only by renaming elements.
Diagnostic: Has the equivalence quotient been applied?
T3 — Local Restrictions versus Global Count. Sylow and solvability results narrow possible structures but may not determine an exact total.
Diagnostic: Which extensions and actions remain to classify?
Structural–Framed Character¶
Number of Groups of a Given Order is hybrid: structurally an enumeration modulo isomorphism and framed by finite-group axioms, order, factorization, and classification theorems.
Structural Core vs. Domain Accent¶
The core is the cardinality of a quotient set: all group laws on n-element carriers modulo relabeling isomorphism. Group theory supplies prime factorization constraints, cyclic and abelian classifications, Sylow theory, solvability, extensions, computational libraries, and asymptotic estimates.
Instantiates / Related Primes¶
This entry is a kind of Function (Mapping).
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Approved root. Order (Group Theory) provides the input cardinality but is not a necessary genus for the counting function under the frozen edge rules.
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Related — finite group, group order, group isomorphism, SmallGroups library, p-group, solvable group, and group extension. These provide the objects, equivalence, and enumeration tools.
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.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
Not to Be Confused With¶
- Order of a Group. Tell: Order is the cardinality of one group; this abstraction counts group types sharing that cardinality.
- Number of Subgroups. Tell: Subgroup counts depend on one group's internal lattice.
- Finite Simple Group Classification. Tell: Simple groups are building blocks, but most groups counted at an order are not simple.
- Labeled Cayley Tables. Tell: Relabelings can encode the same isomorphism class and must be identified.
- Asymptotic Group Enumeration. Tell: Asymptotic growth estimates need not give the exact value at a specified n.
References¶
- Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Finite_group (revision 1348071277).
- Preserved source candidate: https://www.ams.org/notices/200407/fea-aschbacher.pdf
- Preserved source candidate: https://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Group_Theory/Group_Theory_and_its_Application_to_Chemistry
- Preserved source candidate: http://groupnames.org
- Preserved source candidate: http://www.bluetulip.org/programs/finitegroups.html
The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.