On the Baer–Lovász–Tutte construction of groups from graphs: Isomorphism types and homomorphism notions

He, Xiaoyu; Qiao, Youming

doi:10.1016/j.ejc.2021.103404

Cited by 6 publications

(4 citation statements)

References 10 publications

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“…It follows that for an odd prime 𝑝, 𝐇 Γ (𝐅 𝑝 ) is isomorphic to the finite 𝑝-group attached to Γ by Li and Qiao [17]. He and Qiao [13,Thm 1.1] showed that for graphs Γ and Γ ′ and an odd prime 𝑝, 𝐇 Γ (𝐅 𝑝 ) and 𝐇 Γ ′ (𝐅 𝑝 ) are isomorphic if and only if Γ and Γ ′ are. Proof.…”

Section: Centralisers In Graphical Groups and Graphical Lie Algebrasmentioning

confidence: 99%

“…It follows that for an odd prime

p

\mathbf {H}_{\Gamma} (\mathbf {F}_p)

is isomorphic to the finite

p

‐group attached to

\Gamma

by Li and Qiao [17]. He and Qiao [13, Thm 1.1] showed that for graphs

\Gamma

and

\Gamma ^{\prime }

and an odd prime

p

\mathbf {H}_{\Gamma} (\mathbf {F}_p)

and

\mathbf {H}_{\Gamma ^{\prime }}(\mathbf {F}_p)

are isomorphic if and only if

\Gamma

and

\Gamma ^{\prime }

are. Proposition (i)The group centraliser of

h \in R\, \Gamma

\mathbf {H}_{\Gamma} (R)

and the Lie centraliser of

h

h_{Γ} (R...

…”

Section: Graphical Groups and Group Schemesmentioning

confidence: 99%

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Enumerating conjugacy classes of graphical groups over finite fields

Rossmann

2022

Bulletin of London Math Soc

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Section: Centralisers In Graphical Groups and Graphical Lie Algebrasmentioning

confidence: 99%

“…It follows that for an odd prime

p

\mathbf {H}_{\Gamma} (\mathbf {F}_p)

is isomorphic to the finite

p

‐group attached to

\Gamma

by Li and Qiao [17]. He and Qiao [13, Thm 1.1] showed that for graphs

\Gamma

and

\Gamma ^{\prime }

and an odd prime

p

\mathbf {H}_{\Gamma} (\mathbf {F}_p)

and

\mathbf {H}_{\Gamma ^{\prime }}(\mathbf {F}_p)

are isomorphic if and only if

\Gamma

and

\Gamma ^{\prime }

are. Proposition (i)The group centraliser of

h \in R\, \Gamma

\mathbf {H}_{\Gamma} (R)

and the Lie centraliser of

h

h_{Γ} (R...

…”

Section: Graphical Groups and Group Schemesmentioning

confidence: 99%

Enumerating conjugacy classes of graphical groups over finite fields

Rossmann

2022

Bulletin of London Math Soc

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“…Key motivation for GPI is due to its close relation to GI. In the Cayley (verbose) model, GPI reduces to GI [71], while GI reduces to the succinct GPI problem [40,58] (recently simplified [39]). In light of Babai's breakthrough result that GI is quasipolynomial-time solvable [4], GPI in the Cayley model is a key barrier to improving the complexity of GI.…”

Section: Introductionmentioning

confidence: 99%

On the Descriptive Complexity of Groups without Abelian Normal Subgroups (Extended Abstract)

Grochow,

Levet

2023

Electron. Proc. Theor. Comput. Sci.

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In this paper, we explore the descriptive complexity theory of finite groups by examining the power of the second Ehrenfeucht-Fraïssé bijective pebble game in Hella's (Ann. Pure Appl. Log., 1989) hierarchy. This is a Spoiler-Duplicator game in which Spoiler can place up to two pebbles each round. While it trivially solves graph isomorphism, it may be nontrivial for finite groups, and other ternary relational structures. We first provide a novel generalization of Weisfeiler-Leman (WL) coloring, which we call 2-ary WL. We then show that the 2-ary WL is equivalent to the second Ehrenfeucht-Fraïssé bijective pebble game in Hella's hierarchy.Our main result is that, in the pebble game characterization, only O(1) pebbles and O(1) rounds are sufficient to identify all groups without Abelian normal subgroups (a class of groups for which isomorphism testing is known to be in P; Babai, Codenotti, & Qiao, ICALP 2012). In particular, we show that within the first few rounds, Spoiler can force Duplicator to select an isomorphism between two such groups at each subsequent round. By Hella's results (ibid.), this is equivalent to saying that these groups are identified by formulas in first-order logic with generalized 2-ary quantifiers, using only O(1) variables and O(1) quantifier depth.

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“…The Group Isomorphism problem is closely related to the Graph Isomorphism problem (GI). In the Cayley (verbose) model, GpI reduces to GI [ZKT85], while GI reduces to the succinct GpI problem [HL74,Mek81] (recently simplified [HQ21]). In light of Babai's breakthrough result that GI is quasipolynomialtime solvable [Bab16], GpI in the Cayley model is a key barrier to improving the complexity of GI.…”

Section: Introductionmentioning

confidence: 99%

On the parallel complexity of Group Isomorphism via Weisfeiler-Leman

Grochow¹,

Levet²

2021

Preprint

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In this paper, we show that the constant-dimensional Weisfeiler-Leman algorithm for groups (Brachter & Schweitzer, LICS 2020) serves as a unifying isomorphism test for several families of groups that were previously shown to be in P by multiple methods. This includes coprime extensions where the normal Hall subgroup is Abelian and the complement is O(1)-generated (Qiao, Sarma, & Tang, STACS 2011), and groups with no Abelian normal subgroups (Babai, Codenotti, & Qiao, ICALP 2012). In both of these cases, the previous strategy involved identifying key group-theoretic structure that could then be leveraged algorithmically, resulting in different algorithms for each family. A common theme among these is that the group-theoretic structure is mostly about the action of one group on another. Our main contribution is to show that a single, combinatorial algorithm (Weisfeiler-Leman) can identify those same group-theoretic structures in polynomial time.We also derive consequences for the parallel complexity of testing isomorphism in these families. In particular, we show that constant-dimensional Weisfeiler-Leman only requires a constant number of rounds to identify groups in the above classes. Combining this result with the parallel Weisfeiler-Leman implementation of Grohe & Verbitsky for graphs (ICALP 2006), this puts isomorphism testing for each of these families of groups into TC 0 . However, we show that count-free Weisfeiler-Leman is not strong enough to put isomorphism testing even for Abelian groups into AC 0 . * ML thanks Keith Kearnes for helpful discussions, which led to a better understanding of the Hella-style pebble game. ML also wishes to thank Richard Lipton for helpful discussions regarding previous results. JAG was partially supported by NSF award DMS-1750319 and NSF CAREER award CCF-2047756 and during this work. ML was partially supported by J. Grochow startup funds.

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On the Baer–Lovász–Tutte construction of groups from graphs: Isomorphism types and homomorphism notions

Cited by 6 publications

References 10 publications

Enumerating conjugacy classes of graphical groups over finite fields

Enumerating conjugacy classes of graphical groups over finite fields

On the Descriptive Complexity of Groups without Abelian Normal Subgroups (Extended Abstract)

On the parallel complexity of Group Isomorphism via Weisfeiler-Leman

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