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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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    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".2011-06-17

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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 point2011-06-18
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"A discrete group is a group equipped with the discrete topology." http://en.wikipedia.org/wiki/Discrete_group

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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    @user6312, good point.2011-06-17
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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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    @pal$i$o: yes. Any set can be given the discrete topology, and any group can be given the discrete topology to ma$k$e $i$t a discrete group.2011-06-17
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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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In my experience the term discrete group can mean different things in different contexts, and in some contexts it can be slightly ambiguous.
(1) If $G$ is a subgroup of a topological group, then being discrete means that $G$ with its subspace topology is discrete.
(2) If not in case (1), but $G$ acts on a metric or topological space $X$, then being discrete means that all the orbits in $X$ are discrete. If some algebraic property of $G$ is being deduced from the action then you probably want the action to be faithful as well.
(3) If no topology or action is mentioned, then discrete likely means finitely generated.

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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.