Justin Lynd

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math.GR2026

Binary partial groups

Philip Hackney, Justin Lynd, Edoardo Salati

There are many examples of `binary' partial groups in the literature: sets equipped an identity and a partially-defined binary operation, such that each element admits an inverse.…

math.GR2026

Embeddable partial groups

Philip Hackney, Justin Lynd, Edoardo Salati

We record a folklore theorem that says a partial group embeds in a group if and only if each word has at most one possible multiplication, regardless of choice of parenthesization.…

math.GR2026

Splitting the center of a Sylow subgroup

George Glauberman, Justin Lynd

Suppose $p$ is a prime and $S$ is a Sylow $p$ -subgroup of a finite group $G$ . If $S$ is normal in $G$ , then $Z (S)$ is the direct product of $S \cap Z (G)$ with $[Z (S), G]$ . We prove…

math.GR2025

Higher Segal spaces and partial groups

Philip Hackney, Justin Lynd

The d-Segal conditions of Dyckerhoff and Kapranov are exactness properties for simplicial objects based on the geometry of cyclic polytopes in d-dimensional Euclidean space. 2-Sega…

math.GR2025

Components and realizability of fusion systems

Ellen Henke, Justin Lynd

For $p \in {2, 3}$ it is known that a saturated $p$ -fusion system is realizable if and only if each of its components is realizable by a finite simple group. For primes $p \geq 5$ th…

math.GR2025

Cohomology on the centric orbit category of a fusion system

George Glauberman, Justin Lynd

We study here the higher derived limits of mod $p$ cohomology on the centric orbit category of a saturated fusion system on a finite $p$ -group. It is an open problem whether all su…