Coset posets of infinite groups

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 ℋ 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.

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

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 ℋ 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:

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

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.

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 ℋ 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 ⁢ ( ℋ small ⁢ ( G ) ) onto this link.

The vertices from ℋ 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.

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 ⁢ ( ℋ 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.

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