Partial Groups in Algebra and Fusion Systems

• Partial groups are algebraic structures characterized by a partially defined multiplication on words, with unit, inversion, and generalized associativity axioms.
• They are equivalently described as reduced spiny symmetric sets, linking classical group theory to simplicial methods and fusion system frameworks.
• The theory features a robust cohomology and extension framework alongside explicit computational classifications for finite partial groups.

Searching arXiv for recent and foundational papers on partial groups to ground the article and citations. Partial groups are algebraic structures, introduced by Chermak, in which multiplication is assigned not to all finite words in a set but only to a specified domain of words, while still satisfying unit, inversion, and generalized associativity axioms. In contemporary treatments they are equally described as reduced spiny symmetric sets, so the subject sits simultaneously in algebra, simplicial theory, and the theory of fusion systems and localities (Chermak, 2015, Hackney et al., 2023, Hackney et al., 2024).

1. Algebraic definition and immediate structure

In Chermak’s formulation, a partial group consists of a nonempty set $M$, a subset $D\subseteq W(M)$ of the free monoid on $M$, a product map $\Pi:D\to M$, and an involutory bijection $x\mapsto x^{-1}$, extended to words by reversal, such that $M\subseteq D$, $u\circ v\in D\Rightarrow u,v\in D$, $\Pi$ restricts to the identity on one-letter words, and whenever $u\circ v\circ w\in D$, one has $u\circ \Pi(v)\circ w\in D$ together with

$D\subseteq W(M)$0

If $D\subseteq W(M)$1, then $D\subseteq W(M)$2 and $D\subseteq W(M)$3, where $D\subseteq W(M)$4. A group is precisely a partial group with total domain $D\subseteq W(M)$5 (Chermak, 2015).

This formulation encodes a partially defined multiplication on words of arbitrary length rather than merely a partially defined binary law. Basic constructions already resemble ordinary group theory. For $D\subseteq W(M)$6, the domain of conjugation by $D\subseteq W(M)$7 is

$D\subseteq W(M)$8

and for $D\subseteq W(M)$9 one defines

$M$0

A nonempty subset $M$1 is a partial subgroup if it is closed under inversion and under products of words in $M$2. A partial subgroup $M$3 is partial normal if $M$4, $M$5, and $M$6 imply $M$7 (Chermak, 2015).

Chermak further isolates objective partial groups and localities. An objective partial group $M$8 is one whose domain is determined by chains of conjugations among a specified family $M$9 of subgroups. A locality $\Pi:D\to M$0 is a finite objective partial group equipped with a distinguished $\Pi:D\to M$1-subgroup $\Pi:D\to M$2, maximal among $\Pi:D\to M$3-subgroups of $\Pi:D\to M$4. This is the framework in which partial groups became central in the study of fusion systems and transporter systems (Chermak, 2015).

2. Simplicial and symmetric-set formulations

A decisive reformulation identifies partial groups with special symmetric simplicial objects. A symmetric set is a presheaf on finite ordinals and all set maps between them. Such a symmetric set is spiny when its spine maps are injective; in the reduced case, meaning a single $\Pi:D\to M$5-simplex, reduced spiny symmetric sets are equivalent to partial groups, while spiny symmetric sets without reduction are equivalent to partial groupoids (Hackney et al., 2023).

This viewpoint packages admissible words as simplices. If $\Pi:D\to M$6 is the symmetric set associated to a partial group, then $\Pi:D\to M$7 is the underlying set of elements, and an $\Pi:D\to M$8-simplex corresponds to an admissible $\Pi:D\to M$9-word. The injectivity of the relevant Segal-type maps means that a simplex is uniquely determined by its edges. In matrix language, an $x\mapsto x^{-1}$0-simplex $x\mapsto x^{-1}$1 determines entries $x\mapsto x^{-1}$2, and for spiny symmetric sets the assignment

$x\mapsto x^{-1}$3

is injective. This matrix picture is used throughout the modern theory to encode partial composition and inversion (Hackney et al., 2024).

The same formalism sharply separates groups and groupoids from their partial analogues. A symmetric set is isomorphic to the nerve of a groupoid exactly when the spine maps are bijections rather than merely injections. Thus partial groupoids are obtained by weakening the Segal-type bijectivity conditions to injectivity, and partial groups are the one-object case of that weakening (Hackney et al., 2024).

The symmetric-set model also clarifies categorical behavior. The category of spiny symmetric sets is reflective in the category of symmetric sets, and the category of reduced spiny symmetric sets is reflective in reduced symmetric sets. As a consequence, partial groupoids and partial groups are complete and cocomplete; limits are inherited from symmetric sets, while colimits are computed in symmetric sets and then reflected back to the spiny, reduced subcategory (Hackney et al., 2023).

3. Dimension, skeleta, and finiteness

Because symmetric sets admit a robust notion of degeneracy, partial groups carry a natural dimension theory. An element $x\mapsto x^{-1}$4 is degenerate if it factors through a noninvertible surjection $x\mapsto x^{-1}$5 with $x\mapsto x^{-1}$6. The $x\mapsto x^{-1}$7-skeleton $x\mapsto x^{-1}$8 is the smallest symmetric subset containing all simplices of degree at most $x\mapsto x^{-1}$9. One says that $M\subseteq D$0 is $M\subseteq D$1-skeletal if $M\subseteq D$2, and that $M\subseteq D$3 has dimension $M\subseteq D$4 if it is $M\subseteq D$5-skeletal but not $M\subseteq D$6-skeletal (Hackney et al., 2024).

For finite partial groups this dimension is tightly controlled by cardinality. If $M\subseteq D$7 is a partial group with $M\subseteq D$8 elements, regarded as a symmetric set, then $M\subseteq D$9 is $u\circ v\in D\Rightarrow u,v\in D$0-skeletal. If $u\circ v\in D\Rightarrow u,v\in D$1 is actually a group, then its dimension is exactly $u\circ v\in D\Rightarrow u,v\in D$2. More generally, for a connected partial groupoid $u\circ v\in D\Rightarrow u,v\in D$3, if the dimension reaches the maximal value allowed by the number of outgoing nonidentity arrows, then $u\circ v\in D\Rightarrow u,v\in D$4 is already a genuine groupoid. In the one-object case this yields the rigidity statement that maximal possible dimension characterizes groups among partial groups of a given finite size (Hackney et al., 2024).

This has a strong finiteness consequence. Chermak’s original definition uses a subset $u\circ v\in D\Rightarrow u,v\in D$5 of an infinite free monoid, so a finite underlying set can still appear to require infinitely much data. The symmetric-set description shows that this is illusory for finite partial groups: once $u\circ v\in D\Rightarrow u,v\in D$6, all simplices above degree $u\circ v\in D\Rightarrow u,v\in D$7 are degenerate, so the entire structure is determined by finitely many low-dimensional simplices. In particular, finite partial groups have only finitely many im-partial subgroups, meaning nonempty symmetric subsets, and there are only finitely many partial groups up to isomorphism for a fixed finite cardinality (Hackney et al., 2024).

The same dimension theory behaves differently from ordinary simplicial dimension. A finite group $u\circ v\in D\Rightarrow u,v\in D$8 has infinitely many nonempty simplices in its usual simplicial nerve, but when regarded as a symmetric set its dimension is finite, namely $u\circ v\in D\Rightarrow u,v\in D$9. This distinction is one of the conceptual payoffs of the symmetric-set formulation (Hackney et al., 2024).

4. Subgroups, quotients, free objects, and embeddability

The category $\Pi$0 of partial groups admits kernels, cokernels, limits, and colimits. In Salati’s formulation, one distinguishes impartial subgroups, which are precisely images of morphisms in the categorical sense, from partial subgroups, which carry the maximal induced domain. Quotients are defined as cokernels in $\Pi$1, and this construction agrees with ordinary group quotients when the partial group is a group and with Chermak’s quotient of a locality by a partial normal subgroup in the locality setting (Salati, 2021).

The same paper constructs free partial groups. There is a free partial group on a pointed set, and a richer free construction over a category of structured sets that remembers an intended domain of words and an involution. This richer notion yields a generators-and-relations theorem: every partial group is a quotient of a free partial group. In that sense partial groups admit a presentation theory analogous to the classical statement that every group is a quotient of a free group (Salati, 2021).

Embeddability into genuine groups is subtler. A folklore theorem, recorded and extended in a many-object form, states that a partial group embeds in a group if and only if each word has at most one possible multiplication, regardless of choice of parenthesization. Equivalently, all words are “kind” rather than “mean,” in the language of the paper. For partial groupoids, embeddability in a groupoid is equivalent to embeddability of the reduction in a group. The embeddable partial groups and partial groupoids form reflective subcategories, so each partial object has a universal embeddable reflection (Hackney et al., 27 Jan 2026).

At the level of elementary algebra, the category also supports homological analogues of kernels and images. Quotienting by a partial normal subgroup behaves as expected, but the embeddability criterion shows that not every partial group is merely a partial subgroup of some ambient group; ambiguity of word multiplication is the exact obstruction (Hackney et al., 27 Jan 2026).

5. Pregroups, binary partial groups, and related partially defined structures

The modern notion of partial group subsumes older partially defined group-like structures. Stallings’ pregroups admit a canonical embedding into Chermak partial groups: the category of pregroups is a full subcategory of the category of partial groups. For a pregroup $\Pi$2, the associated universal group $\Pi$3 is canonically isomorphic to the fundamental group $\Pi$4 of the simplicial classifying space of the corresponding partial group. This identifies pregroups as a combinatorial subclass of partial groups particularly well adapted to graphs of groups and amalgam-type constructions (Lemoine et al., 2023).

This comparison has consequences for fusion systems. The same work proves that every fusion system on a finite $\Pi$5-group is realizable as the fusion system of some finite pregroup, and that for a Sylow $\Pi$6-subgroup $\Pi$7 the fusion systems $\Pi$8 and $\Pi$9 coincide. This places pregroups and partial groups in the same realizability landscape, though localities remain a much more restrictive class (Lemoine et al., 2023).

A second comparison concerns binary partial groups, meaning unital partial magmas with a partially defined binary product and inverse map satisfying strong cancellation axioms. Many classical examples of “binary” partial groups in the literature turn out to lie inside Chermak’s framework. There is a fully faithful functor from binary partial groups to Chermak partial groups, and the category of binary partial groups is equivalent to the category of $u\circ v\circ w\in D$0-skeletal partial groups. Thus the one- and two-dimensional simplicial data already recover exactly the largest class of binary partial groups that can be canonically regarded as partial groups in Chermak’s sense (Hackney et al., 3 Mar 2026).

These comparison theorems sharpen the conceptual position of partial groups. They are broader than pregroups and binary partial magmas, but low-dimensional truncations recover those older notions exactly when the relevant coherence conditions are present (Lemoine et al., 2023, Hackney et al., 3 Mar 2026).

6. Cohomology and extension theory

A systematic cohomology theory for partial groups can be formulated in two parallel ways. Algebraically, if a partial group $u\circ v\circ w\in D$1 acts on an abelian group $u\circ v\circ w\in D$2 via a homomorphism $u\circ v\circ w\in D$3, one sets

$u\circ v\circ w\in D$4

and defines

$u\circ v\circ w\in D$5

These coboundaries satisfy $u\circ v\circ w\in D$6, giving cohomology groups $u\circ v\circ w\in D$7. Simplicially, one can instead take cohomology of the reduced simplicial set underlying $u\circ v\circ w\in D$8 with coefficients in a local system $u\circ v\circ w\in D$9. The two constructions coincide: $u\circ \Pi(v)\circ w\in D$0 This extends ordinary group cohomology, since for $u\circ \Pi(v)\circ w\in D$1 one recovers the classical bar complex and classical $u\circ \Pi(v)\circ w\in D$2 (Pfammatter, 5 Dec 2025).

The main application is extension theory. An extension of a partial group $u\circ \Pi(v)\circ w\in D$3 by a partial group $u\circ \Pi(v)\circ w\in D$4 is a fibre bundle of simplicial sets $u\circ \Pi(v)\circ w\in D$5 whose total space is again a partial group. Such extensions are encoded by twisting pairs $u\circ \Pi(v)\circ w\in D$6, where $u\circ \Pi(v)\circ w\in D$7 and $u\circ \Pi(v)\circ w\in D$8 satisfy a cocycle condition. Fixing an outer action $u\circ \Pi(v)\circ w\in D$9, one obtains an obstruction class

$D\subseteq W(M)$00

An extension with outer action $D\subseteq W(M)$01 exists exactly when this class vanishes, and when it does, the set of equivalence classes of extensions is a principal homogeneous space for

$D\subseteq W(M)$02

For groups this specializes to the Eilenberg–MacLane classification of group extensions (Pfammatter, 5 Dec 2025).

The theory also yields explicit enumerative results. If $D\subseteq W(M)$03 and $D\subseteq W(M)$04 are finite pointed sets with free partial groups $D\subseteq W(M)$05 and $D\subseteq W(M)$06, then the number of equivalence classes of extensions of $D\subseteq W(M)$07 by $D\subseteq W(M)$08 is

$D\subseteq W(M)$09

This contrasts sharply with free groups, whose extension theory is typically infinite (Pfammatter, 5 Dec 2025).

7. Localities, examples, and the computational landscape

Partial groups were introduced to recast transporter and fusion-theoretic structures in more group-like terms. In Chermak’s locality theory, finite localities carry distinguished $D\subseteq W(M)$10-subgroups and object sets controlling which words are defined, normalizers of objects are honest groups, and quotients by partial normal subgroups preserve the locality structure. This places partial groups at the algebraic core of one of the main approaches to saturated fusion systems (Chermak, 2015).

The theory also supports more combinatorial constructions. For every simple graph $D\subseteq W(M)$11, there is a path partial group $D\subseteq W(M)$12 whose automorphism group satisfies

$D\subseteq W(M)$13

Consequently, for any abstract group $D\subseteq W(M)$14 there exist infinitely many nonisomorphic partial groups $D\subseteq W(M)$15 with

$D\subseteq W(M)$16

This universality phenomenon has no analogue for ordinary groups and shows that partial groups are flexible enough to encode arbitrary prescribed automorphism groups (Ramos et al., 2021).

A different line of work studies finite partial groups by exhaustive computation. There are $D\subseteq W(M)$17 partial groups of order at most $D\subseteq W(M)$18 and $D\subseteq W(M)$19 partial groups of order $D\subseteq W(M)$20. The same program gives a complete list of indecomposable partial groups of order at most $D\subseteq W(M)$21, and proves structural results first observed experimentally, including the characterization of indecomposable partial groups of order $D\subseteq W(M)$22 and dimension $D\subseteq W(M)$23 as $D\subseteq W(M)$24 for a unique group $D\subseteq W(M)$25, and the theorem that a spiny symmetric set of degree at most $D\subseteq W(M)$26 is $D\subseteq W(M)$27-coskeletal (Hackney, 25 May 2026).

Taken together, these developments show that partial groups form a broad algebraic class with several simultaneous faces: they are generalized groups defined by partial word multiplication, reduced spiny symmetric sets, algebraic models for localities and fusion data, receptacles for pregroups and binary partial groups, and objects with a genuine cohomology and extension theory. The subject now includes structural rigidity results, categorical generators-and-relations techniques, embeddability criteria, explicit universal constructions, and large-scale computational classification (Hackney et al., 2023, Hackney et al., 2024, Pfammatter, 5 Dec 2025).