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What, exactly, is a discrete group?

In my understanding, a discrete group is a group $G$ on which the only topology that can be given is the discrete topology. For example, the group $S^1$ is not discrete because we can give it the topology inherited from $\mathbb C$.

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There is no set on which the only topology that can be given is discrete. You can always put the indiscrete topology as well. – Joe Johnson 126 Jun 17 '11 at 14:00
@Joe, for the empty set and singletons the discrete topology and the indiscrete topology coincide. – lhf Jun 17 '11 at 14:05
@Ihf: Thanks. I forgot about the trivial examples. – Joe Johnson 126 Jun 17 '11 at 17:56
Often, "discrete" is used more to say "we don't care about the topology on this group", more than "we're going to put a specific topology on this group". – MartianInvader Jun 17 '11 at 20:30
up vote 9 down vote accepted

In the setting in which the phrase would be used, $G$ is not simply a group, but a topological group. A discrete group is a topological group in which the topology is discrete.

For example, let us look at the reals under addition, but equip the reals with the discrete topology. This gives us a topological group, which by definition is discrete.

The fact that the reals can be equipped with a non-discrete topology (such as the usual one) which is compatible with addition is not relevant.

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exactly my point – ncmathsadist Jun 18 '11 at 4:37

"A discrete group is a group equipped with the discrete topology."

If a set has more than one element then it can be given a non-discrete topology and so it does not make sense to require that "the only topology that can be given is the discrete topology".

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The condition that the topology be Hausdorff is often built into the definition of topological group. If that is done, a finite group is always discrete. – André Nicolas Jun 17 '11 at 13:56
@user6312, good point. – lhf Jun 17 '11 at 14:03

The term 'discrete' seems to be applied to a topology here. The unit circle with the euclidean topology is a different topological group from the unit circle with the discretee topology.

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you are saying that they are the same as groups but different as topological spaces? for exmaple the number of path components of $S^1$ is 1 as we can see on a picture but you are saying that we can make it discrete and the circle will have a number of path components that is infinite $|S^1|$? – palio Jun 17 '11 at 14:35
@palio: yes. Any set can be given the discrete topology, and any group can be given the discrete topology to make it a discrete group. – Qiaochu Yuan Jun 17 '11 at 17:26

If $(G, \tau)$ is a topological group. Then, G is a discrete topological group if $\tau$ is the discrete topology on G.

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The point here is as I see it that in most of cases, the term is used rather for subgroups. We call a subgroup $H$ of a topological group $G$ discrete if the induced topology on $H$ is the discrete one. Also, when one refers to a discrete group it very often means that that the group in question is embeddedable naturally in some topological group (whose topology is quite indiscrete) wherein the former group is a discrete subgroup.

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