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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.

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    $\begingroup$ A group with indiscrete topology. $\endgroup$ – Crostul Dec 6 '14 at 11:31
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    $\begingroup$ Every finite Hausdorff space is discrete. A connected finite topological group is indiscrete. $\endgroup$ – Daniel Fischer Dec 6 '14 at 11:31
  • $\begingroup$ Each $T_0$ topological group is Tychonoff. $\endgroup$ – Alex Ravsky Dec 8 '14 at 5:01
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$G=\{0,1,2,3\}$ , $\tau=\{\emptyset, G, \{0,2\},\{1,3\}\}$. Then $(G,+_4,\tau)$ is disconnect toplogical group.

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    $\begingroup$ The question asks for a connected topological group. $\endgroup$ – Daniel Fischer Dec 19 '14 at 13:24
  • $\begingroup$ And a non-abelian group. $\endgroup$ – James Mitchell Jan 1 '15 at 20:57
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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.

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