The most abstract way of defining a group is to use a [group presentation][1]. This is a set of distinct identified group elements (known as generators) together with a set of relations between these generators. Every group has a presentation (but different presentations may define the same group).

For example, the cyclic group with $n$ elements is defined by taking a single generator $a$ and applying the relation $a^n=1$. This is notated as $\langle a | a^n=1 \rangle$ (sometimes the "$=1$" part in relations is taken as read and omitted). The dihedral group with $2n$ elements has two generators $a$ and $b$ and presentation $\langle a,b| a^n=1, b^2=1, (ab)^2=1 \rangle$.

To avoid ambiguity, we say that a presentation defines the *largest possible group* that satisfies the given relations - so the presentation of the cyclic group $C_2$ is $\langle a|a^2=1\rangle$, even though $C_2$ also satisfies $\langle a|a^{2m}=1\rangle$ for any integer $m$.

A group presentation defines the rules that govern composition of the group's elements in the most abstract way, and does not rely on any representation or realisation of the group.


  [1]: https://en.wikipedia.org/wiki/Presentation_of_a_group