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Published by De Gruyter March 10, 2020

# Coset posets of infinite groups

• Kai-Uwe Bux and Cora Welsch
From the journal Journal of Group Theory

## Abstract

We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible geometric realizations.

Communicated by Dessislava H. Kochloukova

Award Identifier / Grant number: SPP 2026

Award Identifier / Grant number: SFB 878

Funding statement: Financial support by the DFG through the programs SPP 2026 and SFB 878 is gratefully acknowledged.

## A Appendix: The coset nerve for proper normal subgroups in a finite group

In this appendix, we consider the family

nor ( G ) : = { N G N G }

of proper normal subgroups of G. Let 𝒩 𝒞 nor ( G ) denote the coset nerve associate to nor ( G ) . This section is devoted to a proof of the following analogue to the result of Shareshian–Woodroofe [4].

## Theorem 20.

If G is finite, N C nor ( G ) is not contractible.

## Proof.

If the group G is trivial, the family of proper normal subgroups is empty. Hence, we assume that G is nontrivial.

The family nor ( G ) is closed with respect to intersections. Hence, we shall consider the associated order complex 𝒫 𝒞 nor ( G ) instead of the homotopy equivalent coset nerve. Our argument uses many ideas of Brown [2, Section 8].

Inducting on the complexity of G, we shall show that 𝒫 𝒞 nor ( G ) is a nontrivial wedge of spheres. Simple groups make up the base of the induction. Fortunately, simple groups are easily understood since the only proper normal subgroup is the trivial subgroup.

## Claim 21.

If G is simple, P C nor ( G ) is a finite discrete set, which we shall identify with G itself. It is 0-spherical and not contractible (since G is nontrivial).

Now assume that G is not simple. Let M G be a minimal normal subgroup. Let π : G G / M denote the canonical projection.

## Claim 22.

Assume that G contains no proper normal subgroup that surjects onto the quotient G / M . Then the map

q : 𝒫 𝒞 nor ( G ) 𝒫 𝒞 nor ( G / M ) , g N π ( g N )

is a homotopy equivalence. Thus, sphericity and non-contractibility of P C nor ( G ) is inherited from P C nor ( G / M ) .

To see this, consider the intersection-closed family 𝒦 : = { N G M N } . The elements of 𝒦 are in 1–1 correspondence to the proper normal subgroups of G / M . Therefore, 𝒫 𝒞 𝒦 ( G ) and 𝒫 𝒞 nor ( G / M ) are isomorphic. On the other hand, 𝒦 is co-final in nor ( G ) since by hypothesis NM is a proper normal subgroup of G for any proper normal N G . This proves Claim 22.

It remains to deal with the case that there is a proper normal subgroup N ~ G that surjects onto G / M . Note that the intersection N ~ M is normal in G. As N ~ is proper, but the product N ~ M is all of G, we conclude that N ~ M M . As M is a minimal normal subgroup in G, we find N ~ M = { 1 } and G = M × N ~ . In particular, we can identify N ~ with G / M . We call a coset gH in G large if it surjects onto N ~ . We call it small otherwise. Note that the coset gH is small if and only if 1 H is small.

We consider the following families:

large ( G ) : = { N G 1 N is large } ,
small ( G ) : = { N G 1 N is small } ,
: = { M × L L N ~ } .

Note that small ( G ) and are closed with respect to intersections. Moreover, small ( G ) is co-final in small ( G ) . More precisely, for N small ( G ) , we have N M × π N ~ ( N ) . Hence, we have the homotopy equivalence

(A.1) 𝒫 𝒞 ( small ( G ) ) 𝒫 𝒞 ( ) 𝒫 𝒞 nor ( N ~ ) .

Now consider a large normal subgroup L G . We claim that the projection onto N ~ restricts to an isomorphism on L. The reason is that M is minimal, and because of this, L M is trivial or all of M. However, the latter possibility is excluded since L is a proper subgroup of G. This argument shows the following.

## Claim 23.

If L is a large proper normal subgroup of G, then G = M × L .

Consequently, all large proper normal subgroups have the same cardinality. Thus, they are mutually incomparable with respect to inclusion.

We now see the structure of the coset poset 𝒫 𝒞 nor ( G ) . Vertices from small ( G ) form the bottom part whose order complex is homotopy equivalent to 𝒫 𝒞 nor ( N ~ ) . Even more is true. The coset 1 N ~ is large. Its link is spanned by small cosets contained in 1 N ~ . Therefore, this link is isomorphic to 𝒫 𝒞 nor ( N ~ ) , and homotopy equivalence (A.1) is a deformation retraction of the bottom part 𝒫 𝒞 ( small ( G ) ) onto this link.

The vertices from large ( G ) lie above the bottom part. The order complex 𝒫 𝒞 nor ( G ) is the union of the bottom part and the stars of the large vertices. Each such star is just the cone over the link of the vertex, and we may consider these stars independently because two large vertices are never joined by an edge. Moreover, the link of a large coset gL is its descending link, i.e., all vertices in the link are cosets that are contained in gL.

## Claim 24.

The link of a large vertex gL is isomorphic to P C nor ( L ) .

This just follows from the fact that a coset gN is contained in gL if and only if N L .

Now we build 𝒫 𝒞 nor ( G ) by adding the stars of large vertices, one by one, to the bottom part 𝒫 𝒞 ( small ( G ) ) . Adding 1 N ~ as the first large vertex, we cone of its descending link. At this point, we obtain a contractible space. Adding each of the remaining large vertices (and there are at least the other cosets of N ~ ) amounts to wedging on the suspension of its link, i.e., wedging on a copy of Σ ( 𝒫 𝒞 nor ( N ~ ) ) . Thus, we have argued the following.

## Claim 25.

Assume that there is a proper normal subgroup N ~ G that surjects onto G / M . Then P C nor ( G ) is homotopy equivalent to a nontrivial wedge of copies of the suspension Σ ( P C nor ( N ~ ) ) .

Note that the theorem follows by induction from Claims 21, 22, and 25. ∎

## Acknowledgements

We would like to thank Russ Woodroofe for suggesting that a proof of Theorem 20 could be carried out using ideas from Brown [2] rather than following the much steeper path of Shareshian–Woodroofe [4]. We also thank Benjamin Brück, Linus Kramer and Russ Woodroofe for helpful comments on a preliminary version of this paper.

## References

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[4] J. Shareshian and R. Woodroofe, Order complexes of coset posets of finite groups are not contractible, Adv. Math. 291 (2016), 758–773. 10.1016/j.aim.2015.10.018Search in Google Scholar

[5] C. Welsch, On coset posets, nerve complexes and subgroup graphs of finitely generated groups, PhD Thesis, Münster, 2018. Search in Google Scholar

Received: 2019-11-09
Revised: 2020-01-30
Published Online: 2020-03-10
Published in Print: 2020-07-01

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