What is, exactly, a discrete group?

group-theory

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.

"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".

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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