An example of finite, connected topological group

Question

A finite Hausdorff topological group, has discrete topology and every discrete group is totally disconnected. I look for an example of a non-Abelian, finite, connected non-Hausdorff group . I think every non-Abelian, finite, connected groups must be discrete.

Answer 1

$G=\{0,1,2,3\}$ , $\tau=\{\emptyset, G, \{0,2\},\{1,3\}\}$. Then $(G,+_4,\tau)$ is disconnect toplogical group.

Answer 2

Every topological group is completely regular, as the topology is uniformizable. Now every countable connected completely regular space $X$ must be indiscrete: If $x$ is a point outside of the closed set $A$, then some map $f:X\to I$ sends $A$ to $0$ and $x$ to $1$, so by connectedness $f(X)=I$, implying that $X$ is uncountable. Hence any finite connected topological group is indiscrete.

Now just take any non-Abelian group, like $D_3$, and equip it with the indiscrete topology.