## 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.

**Funding source: **Deutsche Forschungsgemeinschaft

**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

of proper normal subgroups of *G*.
Let

## Theorem 20.

*If G is finite,
*

## Proof.

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

The family

Inducting on the complexity of *G*, we shall show that

## Claim 21.

*If G is simple,
*

Now assume that *G* is not simple.
Let

## Claim 22.

*Assume that G contains no proper normal subgroup that surjects onto the quotient
*

*is a homotopy equivalence.
Thus, sphericity and non-contractibility of
*

To see this, consider the intersection-closed family
*NM* is a proper normal subgroup of *G* for any proper normal

It remains to deal with the case that there is a proper normal subgroup
*G*.
As
*G*, we conclude that
*M* is a minimal normal subgroup in *G*, we find
*gH* in *G*
*large* if it surjects onto
*small* otherwise.
Note that the coset *gH* is small if and only if

We consider the following families:

Note that

Now consider a large normal subgroup
*L*.
The reason is that *M* is minimal, and because of this,
*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
*

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
*bottom part* whose order complex is homotopy equivalent to

The vertices from
*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
*

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

Now we build

## Claim 25.

*Assume that there is a proper normal subgroup
*

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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**Received:**2019-11-09

**Revised:**2020-01-30

**Published Online:**2020-03-10

**Published in Print:**2020-07-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston