## Algebra & Number Theory

### The abelian monoid of fusion-stable finite sets is free

Sune Reeh

#### Abstract

We show that the abelian monoid of isomorphism classes of $G$-stable finite $S$-sets is free for a finite group $G$ with Sylow $p$-subgroup $S$; here a finite $S$-set is called $G$-stable if it has isomorphic restrictions to $G$-conjugate subgroups of $S$. These $G$-stable $S$-sets are of interest, e.g., in homotopy theory. We prove freeness by constructing an explicit (but somewhat nonobvious) basis, whose elements are in one-to-one correspondence with the $G$-conjugacy classes of subgroups in $S$. As a central tool of independent interest, we give a detailed description of the embedding of the Burnside ring for a saturated fusion system into its associated ghost ring.

#### Article information

Source
Algebra Number Theory, Volume 9, Number 10 (2015), 2303-2324.

Dates
Received: 3 December 2014
Revised: 31 August 2015
Accepted: 8 October 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842447

Digital Object Identifier
doi:10.2140/ant.2015.9.2303

Mathematical Reviews number (MathSciNet)
MR3437763

Zentralblatt MATH identifier
1369.20024

#### Citation

Reeh, Sune. The abelian monoid of fusion-stable finite sets is free. Algebra Number Theory 9 (2015), no. 10, 2303--2324. doi:10.2140/ant.2015.9.2303. https://projecteuclid.org/euclid.ant/1510842447

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