How Likely is a Finite Group to be Hausdorff?

This is the third in a series of posts on some fun little toy problems of mine. Today’s question is one I came across while trying to come up with an example of a topological group which is not Hausdorff. Anyone familiar with topological groups knows that such examples are scarce in the literature. This is not without reason: any T0\mathrm{T}_0 group is actually Hausdorff, so non-Hausdorff topological groups are particularly pathological. But how scarce really are the non-Hausdorff topological groups? In other words, how likely is a topological group to be Hausdorff?

See Is Every Set Fieldable? and Classification of Abelian Lie Groups for the previous two posts on this series!

We begin by noting that, as stated, our question is ill-posed. First, we should clarify that we are working with a fixed group GG and considering all possible topologies which turn it into a topological group — which we call “GG-topologies”. That is to say, our question is actually: given GG and some random GG-topology, how likely is it to be Housdorff? However — and this is more important — the meaning of “some random topology” is still unclear.

To make the question precise, we focus on the case where GG is finite. This is because a finite set XX can only be endowed with a finite number of distinct topologies — indeed, the power set P(X)\mathcal{P}(X) is finite. Counting how many topologies there are in terms of the size of XX is a famous open problem and these numbers have only being computed for very small XX. Here is a plot of (the logarithm of) the first 19 of these numbers:

A plot of the logarithm of the number of topologies a set with n elements admit
See the excellent page on these computation at the OEIS website!

What is relevant to us is that this implies a finite group GG admits only a finite number of GG-topologies. We may thus consider a random variable τ{G-topologies}\tau \in \{ G \text{-topologies} \} with uniform distribution and ask: can we compute P(G is Hausdorff)=P(G is Hausdorff under τ)\mathbb{P}(G\ \text{is Hausdorff}) = \mathbb{P}(G\ \text{is Hausdorff under}\ \tau)? This is the question we will attempt to answer in this article. Since any discrete group is Hausdorff, we certainly know

P(G is Hausdorff)1#{G-topologies}.\mathbb{P}(G\ \text{is Hausdorff}) \ge \frac{1}{\#\{G\text{-topologies}\}}.

What is perhaps surprising is that P(G is Hausdorff)\mathbb{P}(G\ \text{is Hausdorff}) is precisely 1/#{G-topologies}1/\#\{G\text{-topologies}\}. In other words, the odds of GG being Hausdorff are as slim as they could possibly be! Indeed, it is a well known fact from the theory of finite topological spaces that any a finite Hausdorff space is discrete, so the only finite Hausdorff groups are the discrete ones. All it’s left is to compute #{G-topologies}\# \{G\text{-topologies}\}. To do so, we investigate the closure {1}τ\overline{ \{ 1 \} }^\tau of 1G1 \in G under a given GG-topology τ\tau.

Since the translations by elements of GG are all homeomorphism, {g}τ=g{1}τ\overline{\{g\}}^\tau = g \cdot \overline{\{1\}}^\tau for all gGg \in G. It is then easy to see that

nm{nm}τ=n{m}τn{1}τ={n}τ{1}τn m \in \overline{\{n m\}}^\tau = n \cdot \overline{\{m\}}^\tau \subset n \cdot \overline{\{1\}}^\tau = \overline{\{n\}}^\tau \subset \overline{\{1\}}^\tau

for any n,m{1}τn, m \in \overline{\{1\}}^\tau. Similarly, since the inversion gg1g \mapsto g^{-1} is a homeomorphism, we find n1{11}τ={1}τn^{-1} \in \overline{\{1^{-1}\}}^\tau = \overline{\{1\}}^\tau for any n{1}τn \in \overline{\{1\}}^\tau. Finally, g{1}τg1={g1g1}τ={1}τg \cdot \overline{\{1\}}^\tau \cdot g^{-1} = \overline{\{g 1 g^{-1}\}}^\tau = \overline{\{1\}}^\tau for all gGg \in G and n{1}τn \in \overline{\{1\}}^\tau. This goes to show {1}τ\overline{\{1\}}^\tau is a normal subgroup of GG.

There is thus a natural map

{G-topologies}{NG}τNτ={1}τ\begin{aligned} \{G\text{-topologies}\} & \to \{N \triangleright G\} \\ \tau & \mapsto N_\tau = \overline{\{1\}}^\tau \end{aligned}

between the space of GG-topologies and the set of normal subgroups of GG. That’s all well and good, but how does any of this helps us to count GG-topologies? Well, the point is that, perhaps surprisingly, this map is actually a bijection.

To see that our map is injective, we remark that the collection of subsets of GG which are closed under a given GG-topology τ\tau is actually determined by NτN_\tau. Indeed,

F=gF{g}τ=gFgNτF = \bigcup_{g \in F} \overline{\{g\}}^\tau = \bigcup_{g \in F} g \cdot N_\tau

for any FGF \subset G which is closed under τ\tau. Conversely, since GG is finite, any union of NτN_\tau-cosets is closed under τ\tau — a finite union of closed sets is closed. All in all, the subsets which are closed under τ\tau are precisely the unions of NτN_\tau-cosets.

This last equation also goes to show that our map is surjective: given some NGN \triangleright G, the set τ={gXgN:XG}\tau = \left\{ \bigcup_{g \in X} g \cdot N : X \subset G \right\} is a natural candidate for a GG-topology such that Nτ=NN_\tau = N. It is clear that τ\tau is a topology and {1}τ=N\overline{\{1\}}^\tau = N. With enough effort, one can also show that if m:G×GGm : G \times G \to G and i:GGi : G \to G are the group multiplication and inversion, respectively, then

m1(gN)=hGhN×h1gNi1(gN)=g1N,\begin{aligned} m^{-1}(g \cdot N) & = \bigcup_{h \in G} h \cdot N \times h^{-1} g \cdot N \\ i^{-1}(g \cdot N) & = g^{-1} \cdot N \end{aligned},

which goes to show that τ\tau is indeed a GG-topology.

Finally, #{G-topologies}=#{NG}\#\{G\text{-topologies}\} = \#\{N \triangleright G\} and thus

P(G is Hausdorff)=1#{NG}.\mathbb{P}(G\ \text{is Hausdorff}) = \frac{1}{\# \{ N \triangleright G \}}.

We conclude this article with a table of the values of P(G is Hausdorff)\mathbb{P}(G\ \text{is Hausdorff}) for some noteworthy finite groups GG, which were computed with the help of the Group Explorer library. I would like to thank my dear friend Eduardo Sodré for his help with the problem. I really hope the ride was as interesting to you as it was us both! 😁

P(G is Hausdorff)\mathbb{P}(G\ \text{is Hausdorff})







SnS_n for n5n \ge 5


DnD_n for even nn

13+τ(n)\frac{1}{3 + \tau(n)}

DnD_n for odd nn

11+τ(n)\frac{1}{1 + \tau(n)}