Algebra & Number Theory

The abelian monoid of fusion-stable finite sets is free

Sune Reeh

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure
Secondary: 20J15: Category of groups 19A22: Frobenius induction, Burnside and representation rings

Fusion systems Burnside rings finite groups


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.

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