# 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 $\mathrm{T}_0$ 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?*

We begin by noting that, as stated, our question is ill-posed. First, we should
clarify that we are working with a fixed group $G$ and considering
all possible topologies which turn it into a topological group — which we
call “$G$-topologies”. That is to say, our question is actually:
*given $G$ and some random $G$-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 $G$ is finite. This is because a finite set $X$ can only be endowed with a finite number of distinct topologies — indeed, the power set $\mathcal{P}(X)$ is finite. Counting how many topologies there are in terms of the size of $X$ is a famous open problem and these numbers have only being computed for very small $X$. Here is a plot of (the logarithm of) the first 19 of these numbers:

*excellent*page on these computation at the OEIS website!

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

What is perhaps surprising is that
$\mathbb{P}(G\ \text{is Hausdorff})$ is *precisely*
$1/\#\{G\text{-topologies}\}$. In other words, the odds of
$G$ 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
$\# \{G\text{-topologies}\}$. To do so, we investigate the closure
$\overline{ \{ 1 \} }^\tau$ of $1 \in G$ under a given
$G$-topology $\tau$.

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

for any $n, m \in \overline{\{1\}}^\tau$. Similarly, since the inversion $g \mapsto g^{-1}$ is a homeomorphism, we find $n^{-1} \in \overline{\{1^{-1}\}}^\tau = \overline{\{1\}}^\tau$ for any $n \in \overline{\{1\}}^\tau$. Finally, $g \cdot \overline{\{1\}}^\tau \cdot g^{-1} = \overline{\{g 1 g^{-1}\}}^\tau = \overline{\{1\}}^\tau$ for all $g \in G$ and $n \in \overline{\{1\}}^\tau$. This goes to show $\overline{\{1\}}^\tau$ is a normal subgroup of $G$.

There is thus a natural map

between the space of $G$-topologies and the set of normal subgroups of $G$. That’s all well and good, but how does any of this helps us to count $G$-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 $G$ which are closed under a given $G$-topology $\tau$ is actually determined by $N_\tau$. Indeed,

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

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

which goes to show that $\tau$ is indeed a $G$-topology.

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

We conclude this article with a table of the values of $\mathbb{P}(G\ \text{is Hausdorff})$ for some noteworthy finite groups $G$, 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! 😁

$G$ |
$\mathbb{P}(G\ \text{is Hausdorff})$ |
---|---|

$Q_4$ |
$1/6$ |

$K_4$ |
$1/5$ |

$S_4$ |
$1/4$ |

$S_n$ for $n \ge 5$ |
$1/3$ |

$D_n$ for even $n$ |
$\frac{1}{3 + \tau(n)}$ |

$D_n$ for odd $n$ |
$\frac{1}{1 + \tau(n)}$ |