Partial Group Symmetry in Figures I: Semidirect Products and the Six Coins

Takahiro Hayashi

REVIEW 4 major objections 4 minor 1 cited by

Partial Group Symmetry in Figures I: Semidirect Products and the Six Coins

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that for figures assembled from congruent or similar translates of one connected component, the partial symmetry group is a semidirect product of a vector-parade partial group with a wedge sum of component symmetry…

desk verdict Solid, honest partial-group framework for figure symmetry; the central quotient theorem survives the stress-test, but the Six Coins example is left uncomputed. read the letter →

arxiv 2506.14304 v2 pith:7Z3SZXGB submitted 2025-06-17 math.GR math.QA

Takahiro Hayashi This is my paper · ORCID

classification math.GRmath.QA

keywords partialgroupssymmetryoffiguressemidirectproductsvectorparadeswedgesumscongruencesonSixCoinssimplicialsets

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper constructs an algebraic object $\mathcal{P}(F)$, the symmetry partial group, for every subset $F$ of $\mathbb{R}^d$, and uses it to describe partial symmetries of disconnected figures. The classical symmetry group $G(F)$ records only motions that send $F$ to itself, while $\mathcal{P}(F)$ records every Euclidean motion that sends one connected component of $F$ onto another, together with how such motions compose along chains of components. The main theorem asserts that when $F$ is made of congruent or similar translated copies of a single connected piece $F_0$, the symmetry partial group is a semidirect product of the vector-parade partial group $V(P)$ with a wedge sum of copies of $G(F_0)$, taken modulo an explicit congruence. The Six Coins is the paper's central example showing that the congruence is necessary, because that figure is not of the trivial type where the unquotiented semidirect product already works. The result makes partial groups a finer and computable invariant of disconnected figures than the usual symmetry group.

What carries the argument

The machinery is the parade construction of a partial group. A parade is a sequence $(\pi_1,\dots,\pi_n)$ of elements of a group $G_0$ for which some starting point $x$ has $x\pi_1,\,x\pi_1\pi_2,\,\dots,\,x\pi_1\cdots\pi_n$ all inside a fixed set $X$; the parade partial group $P(X)=P(X,X_0,G_0)$ collects such sequences and composes them by multiplying consecutive entries. For a figure $F$, $X$ is the set of connected components, $X_0$ is the power set of $\mathbb{R}^d$, and $G_0$ is the Euclidean group, so a length-one element of $\mathcal{P}(F)$ is exactly an isometry moving one component onto another. The paper adds three supporting constructions: partial group actions on sets and on other partial groups, the wedge sum $\bigvee_\lambda G_\lambda$ as the coproduct in the category of partial groups, and a semidirect product $F\ltimes H$ built simplicially and characterized by a universal mapping property. The congruence in (12.8) is the final piece: it identifies two semidirect-product elements precisely when the same rotational symmetry $a$ acts on the displacement between labels in the same way, and this is what makes the description of $\mathcal{P}(F)$ bijective rather than merely surjective.

What would settle it

Take the square analogue of the Six Coins: a $3\times2$ rectangular grid of six congruent, disjoint squares. Enumerate the elements of $\mathcal{P}(F)_1$ directly as the Euclidean isometries that carry one square to another, and compare that set with the right-hand side of (12.6); if the two sets have different sizes, or if two distinct $\sim$-classes are sent to the same isometry, then Theorem 12.3's isomorphism fails for this $F$.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is Theorem 12.3: if $F\subset\mathbb{R}^d$ has connected components $\{F_0+x\mid x\in P\}$ and $G(F_0)\le\mathrm{O}(d)$, then $\mathcal{P}(F)\cong\bigl(V(P)\ltimes\bigvee_{P}G(F_0)\bigr)/{\sim}$, where $V(P)$ is the partial group of vector parades through the label set $P$, $\bigvee_{P}G(F_0)$ is the wedge sum (the coproduct in the category of partial groups) of one copy of $G(F_0)$ per label, and $\sim$ is the congruence $(y-x)\ltimes[y,a]\sim(y'-x')\ltimes[y',a']$ iff $a=a'$ and $(x'-x)a=y'-y$. The same description holds for the special symmetry partial group with $G(F_0)$ replaced by $SG(F_0)$. The quotient is exactly what makes the formula faithful for figures whose global symmetries permute the components while rotating them; the Six Coins, a $3\times2$ array of congruent coins, has a non-translation global symmetry and is therefore not of SDP (semidirect product) type, meaning the unquotiented semidirect product is not already isomorphic to $\mathcal{P}(F)$, so its partial symmetry group cannot be the unquotiented semidirect product.

Load-bearing premise

The main structural theorem assumes the figure is built from translated copies of one connected piece whose internal symmetries involve no translation, and that the arrangement of those copies is faithfully captured by the label set; when a component has its own translational symmetry, or the label set carries extra symmetries as in the Six Coins, the clean isomorphism degrades to a quotient described only by the congruence (12.8), with no simpler normal form supplied.

Editorial extensions

If this is right

For a connected figure $\mathcal{P}(F)=G(F)$, so the new information carried by $\mathcal{P}(F)$ lives entirely in how the disconnected pieces of a figure are arranged.
Two figures can have the same classical symmetry group and the same number of length-one partial symmetries yet have non-isomorphic partial symmetry groups, as shown by the pair with $\mathcal{P}(F)\cong D_3\vee D_4\vee D_4\vee D_5$ and $\mathcal{P}(F')\cong D_3\vee D_3\vee D_4\vee D_6$.
For figures with exactly two congruent components swapped by an isometry, the partial symmetry group is a twisted semidirect product $C_2\ltimes(G(F_+)\vee_{G(F_+)\cap G(F_-)}G(F_-))$, and the twist can be non-trivial, for instance for a pair of prisms related by a rotoreflection.
For two similar but non-congruent components, the similarity partial group satisfies $\mathrm{SimP}(F)\cong V(\{0,1\})\ltimes(G(F_+)\vee_{G(F_+)\cap G(F_-)}G(F_-))$.
When the hypotheses of Proposition 12.4(2) hold, the congruence is harmless and $\mathcal{P}(F)$ is a genuine semidirect product, such as $V(\{x,y,z\})\ltimes(D_3\vee D_3\vee D_3)$ for a three-component figure.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The quotient pattern in (12.8) looks like a cocycle compatibility condition; if it generalizes, partial symmetry groups of repetitive structures such as tilings would be expressible as crossed products of a translation groupoid by a component symmetry group, with the non-SDP cases corresponding to non-trivial cocycles.
The difference between SDP type and genuinely quotiented figures suggests a computable numerical invariant for finite figures: the size or rank of the congruence relation, which measures how much the plain semidirect product overcounts and which may separate figures that agree in $|\mathcal{P}(F)_1|$.
The two-component theorems suggest an algorithmic recognition test: compute $G(F_+)\vee_{G(F_+)\cap G(F_-)}G(F_-)$ together with the twisting action or the vector-parade factor, and compare; Example 7.3 shows that counting $\mathcal{P}(F)_1$ alone is not enough.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I read the paper against the reader's report and the stress-test note. The reader's road map is fair—this is conditional-accept material—but the specific gap flagged in the stress-test does not land. In Theorem 12.3, equality of the maps t_{-x} a t_y and t_{-x'} a' t_{y'} is equivalent to a=a' and (x'−x)a = y'−y, which is exactly (12.8). The proof states this equivalence, and the 'missing' converse is immediate from the explicit form of ψ on generators. So the quotient formula is not half-proved; it is correct.

What is genuinely new: the generalized partial-group action axioms (AP1–AP6), the universal property of the semidirect product (Theorem 11.2), and the classification theorems 12.2 and 12.3, together with the SDP-type criterion in Proposition 12.4. The parade construction is credited to [14] and semidirect products to [2] and [7]; the paper is honest about that and builds on those references in a useful way. The central proofs are explicit and the universal property is a real addition to the partial-group toolkit.

Soft spots are real but minor. Many auxiliary lemmas are left as 'straightforward'—Proposition 6.1(2), Proposition 7.5, parts of Proposition 8.2. That is acceptable for the main results, but a journal referee should ask the authors to expand the ones used later. The 'Six Coins' in the title is anticlimactic: the paper shows the figure is not of SDP type and invokes the quotient, but never computes the quotient. A reader hoping for a concrete description of P(F) for six coins will be disappointed. That is a missed opportunity, not a mathematical flaw. The hypotheses of Theorem 12.3 (G(F0) ≤ O(d)) and Proposition 12.4 are also restrictive; the general case with non-orthogonal stabilizers is left open.

Who is this for? People working on partial groups, localities, and fusion systems, and possibly anyone interested in symmetry invariants of disconnected sets. It is not a breakthrough, but it is technically sound, honest about attribution, and gives a usable bridge between abstract partial-group theory and elementary geometry. I would send it to a referee.

Referee Report

4 major / 4 minor

Summary. The paper develops a partial-group invariant P(F) for subsets F of Euclidean space, based on parades between connected components. It introduces actions of partial groups on sets and on other partial groups, constructs semidirect products through simplicial methods, proves a universal property for these semidirect products, and applies the machinery to figures with two congruent or similar components and to a class containing the Six Coins. The central structural results are Theorem 12.2, for two similar non-congruent components, and Theorem 12.3, for figures whose connected components are translates F0+x of one component, where P(F) is asserted to be isomorphic to a quotient of V(P)⋉∨_P G(F0) by the explicit congruence (12.8). The Six Coins is shown not to be of semidirect-product type.

Significance. The paper makes several genuinely useful contributions: the parade construction P(X), the wedge-sum rigidity theorem (Theorem 7.2), the universal property of the semidirect product (Theorem 11.2), and the worked examples showing how P(F) distinguishes figures that G(F) cannot. The semidirect product construction in Section 10 genuinely extends the Grazian-Henke framework to non-regular partial group actions, and the quotient phenomenon exhibited by the Six Coins is a meaningful new behaviour. The treatment is mostly constructive, with explicit formulas for the maps involved. If Theorem 12.3's explicit quotient description is fully justified, the paper is a solid contribution to the partial-group approach to symmetry. The main weakness is that the proof of the explicit congruence in Theorem 12.3 is incomplete as written, and one stated theorem in Section 12 is left without a proof.

major comments (4)

[§12, Theorem 12.3] The proof of the congruence identification is incomplete. After constructing ψ and applying Proposition 2.2(2), the paper verifies that ψ((y−x)⋉[y,a]) = ψ((y′−x′)⋉[y′,a′]) implies the right-hand side of (12.8), i.e. that the kernel congruence ∼ψ is contained in the displayed relation. To conclude that (V(P)⋉∨_P G(F0))/∼ is isomorphic to P(F), one must also prove the converse: if a=a′ and (x′−x)a = y′−y, then the two elements have the same image under ψ. This converse is true by a short computation using t_u a = a t_{u a} for a∈G(F0)≤O(d), but it is not written. As it stands, (12.8) is only shown to be necessary for the kernel congruence, and the explicit quotient presentation that drives the Six Coins discussion is not established.
[§12, Theorem 12.1] The proof of Theorem 12.1 is omitted with the remark that it is very similar to that of Theorem 12.2. This is a load-bearing gap because Theorem 12.1 is not a corollary of Theorem 12.2: the hypotheses differ (congruent components with no swapping isometry versus non-congruent similar components), and the proof of Theorem 12.2 relies on the diameter assumption and on the decomposition (12.2) that uses non-congruence. The author should either give the full proof or state explicitly which parts of the proof of Theorem 12.2 carry over unchanged and why the special case SP(F) follows.
[§7, Proposition 7.5] Proposition 7.5, which identifies G+∪G− with the pushout G+∨_H G− for H=G+∩G−, is justified by a two-sentence universal-property sketch. This proposition is used in the proofs of Theorem 8.5 and Theorem 12.2, so it is load-bearing. Please provide a complete proof, including well-definedness of the induced map on all n-simplices of the pushout and verification that the map from G+∪G− to the pushout is a map of partial groups.
[§6, §8] Several assertions are delegated to "straightforward" without proof despite being used later, notably Proposition 6.1(2) (P(F)=G(F) for connected F) and Proposition 8.2(4)–(5) (membership of G(X) and G+∩G− in the normalizer). These are plausible and elementary, but since they are invoked in later structural theorems, the proofs should be supplied or at least precisely referenced.

minor comments (4)

[§11, Lemma 11.1] In the proof of Lemma 11.1, the cross-reference "by (6.3)" does not match any displayed equation in the paper; it should refer to an earlier condition in Section 2 or Section 3.
[§12, Theorem 12.3 proof] The displayed line in the final paragraph of the proof, "at(x′−x)a = tx′−xa = a′ty′−y," appears garbled; please correct the notation so that the intended equality of translations and orthogonal parts is clear.
[§12, Theorems 12.2 and 12.3] The letter F is used both for the geometric figure and for the partial group V({0,1}) (or V(P)) inside the proofs; this is confusing and should be resolved by using different letters, for example G for the auxiliary partial group.
[§12, Six Coins] Since the Six Coins is the motivating example, it would strengthen the paper to include at least a schematic description of the quotient in (12.8) for the 3×2 label set P, rather than only proving that the figure is not of SDP type.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's structural theorems are derived from explicit universal-property constructions and direct computations, with external citations used only for standard background facts.

full rationale

The paper's central claims are not circular. The semidirect product in Theorem 11.2 is characterized by a universal property and explicit formulas (11.6), and the application to figures in Theorem 12.3 proceeds by constructing maps via Theorem 11.2, verifying surjectivity with a direct parade argument, and computing the induced congruence with an explicit affine calculation. No parameter is fitted and no prediction is assumed. The description of the congruence in (12.8) is asserted after showing one direction and stating the converse; even if the converse proof is abbreviated, it is a direct algebraic verification and not an input to the theorem. Citations to Chermak, Broto-Gonzalez, Gonzalez, and Grazian-Henke are to external, prior work used for standard partial-group facts (extension theory, normalizers, semidirect-product framework), not for the paper's target results about P(F). The only self-reference is to a forthcoming paper [10], which is not load-bearing. There is no fitting, no renamed known result, and no definition that presupposes the theorem it is used to prove.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces definitions such as F-partial groups and semidirect products, but these are constructed mathematical objects rather than postulated entities. The central claim rests on standard external theorems in partial group theory and on explicit hypotheses about the figure F, with no free parameters fitted to data.

assumptions (5)

standard math The category of partial groups is complete and cocomplete, so wedge sums and fiber sums exist as colimits.
Invoked at the start of Section 7, which cites references [3] and [15] for colimits and uses them to define G+ wedge_H G-.
standard math Broto-Gonzalez extension theory: for a factor set (twist, sigma), the set G semidirect H with the prescribed product is a partial group.
Section 8 uses reference [2, Theorem 4.5] to assert that the twisted semidirect product is a partial group.
standard math The simplicial description of partial groups and prepartial groups is equivalent to the algebraic description.
Sections 9 and 10 rely on references [2] and [6] for the spine operator, inversion, and the associated partial group construction used to define F semidirect H.
standard math Lemma 2.9(a) of reference [2] characterizes the normalizer N(P) and is used to define conjugation actions.
Proposition 8.2 invokes this lemma to establish the normalizer of a partial group and the two-sided action in (8.1).
standard math Standard Lebesgue measure and integrability assumptions underlie the center of gravity computation in Proposition 5.2.
Proposition 5.2 requires F to be Lebesgue measurable with positive finite measure and integrable coordinate functions to define z(F).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Partial Group Symmetry in Figures I: Semidirect Products and the Six Coins." pith.science (2026). https://pith.science/paper/7Z3SZXGB

@misc{pith2026250614304,
  author       = {Pith},
  title        = {Pith review of: Partial Group Symmetry in Figures I: Semidirect Products and the Six Coins},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Z3SZXGB}},
  note         = {Machine review of arXiv:2506.14304}
}

abstract

In this paper, we construct a partial group $\mathcal{P}(F)$ that represents the "partial symmetry" inherent in a subset $F$ of $d$-dimensional Euclidean space. In cases where $F$ is not connected, $\mathcal{P}(F)$ captures more detailed information than the conventional symmetry group $G(F)$. To establish a stronger connection between $\mathcal{P}(F)$ and $F$, we introduce a novel definition of partial group action. Furthermore, to characterize $\mathcal{P}(F)$ in specific cases, we define partial group actions on other partial groups and present a construction of the corresponding semidirect product.

Figures

Figures reproduced from arXiv: 2506.14304 by the authors.

**Figure 1.** Six Coins. gle, it overlooks the distinct rotational symmetry of each individual coin as well as the overall “3 × 2 translation symmetry” present in the figure (refer to Section 5 for our definition of group symmetry). To address these cases of “partial symmetry,” we further develop the framework of symmetry in figures, drawing upon Chermak’s introduction of a class of algebraic objects based on partially defined op… view at source ↗

**Figure 2.** (a) Two disks F. (b) Parametrization of P(F)1. (c) Disk and annulus F ′ . (d) Disk with a component without symmetry, F ′′ . Example 6.2. (1) Let F be the regular tetragon [−1, 1]2 in R 2 . By Proposition 6.1 (2) and Proposition 5.2, we have P(F) ∼= G(F, R 2 , O(2)) ∼= D4. (2) (Two disks of same size) Let F ⊂ R 2 be the union D+ ⨿ D− of two disks D± := D ± e1, where e1 := (1, 0) and D := {(x, y) | x 2 + y 2 ≤ 1/4}.… view at source ↗

**Figure 3.** Configurations F and F ′ . 20 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Two balls of different sizes. Example 7.7. Let F be the disjoint union of two d-dimensional balls of different sizes, namely, Br+ +ed and Br− −ed, where Br± := {x ∈ R d | |x| ≤ r±}, 0 < r− < r+ < 1, and ed := (0, . . . , 0, 1) ∈ R d (cf [PITH_FULL_IMAGE:figures/full_f…

**Figure 5.** Figure 5: Reflection symmetry of two triangles. where A(α) := (η1 ↼ (c2, c3, . . . cn), . . . , ηn−1 ↼ cn, ηn), (8.3) ηi ↼ (ci+1, ci+2, . . . cn) := (· · ·(ηi ↼ ci+1) ↼ · · · ↼ cn−1) ↼ cn (8.4) for each α = ((ci , ηi))i ∈ (G × H1) n . By Broto and Gonzalez [2, Theorem 4.5], G⋉↼,…

**Figure 6.** Figure 6: Two prisms. Example 8.6. Let a ∈ O(3) be the composition of a 90◦ rotation about the z-axis with a reflection across the xy-plane. Consider prisms F+ and F− := F+a with empty intersections such that z(F±) = (0, 0, ±1) and that their bases are parallelograms that are pa…

**Figure 7.** Figure 7: A figure of SDP type. Example 12.5. (1) By Part (1) of the above Proposition, the figure of the Six Coins is not of SDP type. (2) Let P = {x, y, z} ⊂ R 2 and F be as in [PITH_FULL_IMAGE:figures/full_fig_p042_7.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Higher Segal spaces and partial groups
math.GR 2025-07 conditional novelty 7.0 of 10

The higher Segal degree of a partial groupoid equals the Helly number of the closure space of a characteristic action, and the method gives explicit degrees for punctured Weyl groups.

Reference graph

Works this paper leans on

16 extracted references · 12 canonical work pages · cited by 1 Pith paper

[14]

C. Okay, V. Castillo, W. H. Stern, Simplicial effects and weakly asso- ciative partial groups, arXiv:2502.05958, 2025

work page Pith review arXiv 2025
[2]

Broto, A

C. Broto, A. Gonzalez, An extension theory for partial groups, arXiv:2105.03457, 2021

arXiv 2021
[7]

Semidirect Products and Wreath Products of Localities

V. Grazian, E. Henke, Semidirect Products and Wreath Products of Localities, arXiv:1905.03762, 2019

work page Pith review arXiv 1905
[1]

Aschbacher, R

M. Aschbacher, R. Kessar, B. Oliver, Fusion Systems in Algebra and Topology, Lond. Math. Soc. Lecture Note Ser. 391, Cambridge Univ. Press, 2011

work page 2011
[3]

Chermak, Fusion systems and localities, Acta Math

A. Chermak, Fusion systems and localities, Acta Math. 211 (1) (2013), 47-139. https://doi.org/10.1007/s11511-013-0099-5

doi:10.1007/s11511-013-0099-5 2013
[4]

Chermak, Finite localities I, Forum Math

A. Chermak, Finite localities I, Forum Math. Sigma 10 (2022), 1-31. https://doi.org/10.1017/fms.2022.31

work page doi:10.1017/fms.2022.31 2022
[5]

Finite Localities III

A. Chermak, Finite localities III, arXiv:1610.06161, 2016

work page Pith review arXiv 2016
[6]

Gonz´ alez, An extension theory for partial groups and localities, arXiv:1507.04392, 2015

A. Gonz´ alez, An extension theory for partial groups and localities, arXiv:1507.04392, 2015

arXiv 2015

Show all 16 references

[8]

Hackney, Higher Segal spaces and partial groups, in Homotopical Al- gebra and Higher Structures, edited by M

P. Hackney, Higher Segal spaces and partial groups, in Homotopical Al- gebra and Higher Structures, edited by M. Batanin, A. Lazarev, M. Liv- ernet, and M. Markl, Oberwolfach Reports, no. 39 (2024), pp. 2291–

2024
[9]

Hackney, J

P. Hackney, J. Lynd, Partial groups as symmetric simplicial sets, J. Pure Appl. Algebra 229 (2) (2025), 107864. https://doi.org/10. 1016/j.jpaa.2025.107864

2025
[10]

Hayashi, Partial group symmetry in figures II, in preparation

T. Hayashi, Partial group symmetry in figures II, in preparation. 43
[11]

Howie, Fundamentals of semigroup theory, Lond

J. Howie, Fundamentals of semigroup theory, Lond. Math. Soc. Monogr. New Ser. 12, Oxford Univ. Press, London, 1995

1995
[12]

Lynd, Partial groups and higher Segal conditions, in Finite Groups, Fusion Systems and Applications , edited by I

J. Lynd, Partial groups and higher Segal conditions, in Finite Groups, Fusion Systems and Applications , edited by I. Capdeboscq, E. Henke, and M. Liebeck, Oberwolfach Reports, no. 16 (2025), pp. 25-28. https: //doi.org/10.4171/OWR/2025/16

2025 doi
[13]

Martino, S

J. Martino, S. Priddy, Unstable homotopy classification of BG∧ p , Math. Proc. Cambridge Philos. Soc. 119 (1996), 119-137. https://doi.org/ 10.1017/s030500410007403x

1996 doi
[15]

Salati, Limits and colimits, generators and relations of partial groups, J

E. Salati, Limits and colimits, generators and relations of partial groups, J. Algebra. 622 (2023), 291-327. https://doi.org/10.1016/ j.jalgebra.2023.01.014. Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8604, Japan Email address: hayashi@math.nago...

2023
[2294]

https://doi.org/10.4171/OWR/2024/39

2024 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [14]

C. Okay, V. Castillo, W. H. Stern, Simplicial effects and weakly asso- ciative partial groups, arXiv:2502.05958, 2025

work page Pith review arXiv 2025

[2] [2]

Broto, A

C. Broto, A. Gonzalez, An extension theory for partial groups, arXiv:2105.03457, 2021

arXiv 2021

[3] [7]

Semidirect Products and Wreath Products of Localities

V. Grazian, E. Henke, Semidirect Products and Wreath Products of Localities, arXiv:1905.03762, 2019

work page Pith review arXiv 1905

[4] [1]

Aschbacher, R

M. Aschbacher, R. Kessar, B. Oliver, Fusion Systems in Algebra and Topology, Lond. Math. Soc. Lecture Note Ser. 391, Cambridge Univ. Press, 2011

work page 2011

[5] [3]

Chermak, Fusion systems and localities, Acta Math

A. Chermak, Fusion systems and localities, Acta Math. 211 (1) (2013), 47-139. https://doi.org/10.1007/s11511-013-0099-5

doi:10.1007/s11511-013-0099-5 2013

[6] [4]

Chermak, Finite localities I, Forum Math

A. Chermak, Finite localities I, Forum Math. Sigma 10 (2022), 1-31. https://doi.org/10.1017/fms.2022.31

work page doi:10.1017/fms.2022.31 2022

[7] [5]

Finite Localities III

A. Chermak, Finite localities III, arXiv:1610.06161, 2016

work page Pith review arXiv 2016

[8] [6]

Gonz´ alez, An extension theory for partial groups and localities, arXiv:1507.04392, 2015

A. Gonz´ alez, An extension theory for partial groups and localities, arXiv:1507.04392, 2015

arXiv 2015