Free group:F2

Definition
The free group of rank two, also written as $$F_2$$, is defined as the free group on a generating set of size two. $$2$$ is the smallest possible rank for a free non-abelian group (the free groups of rank $$0$$ and $$1$$ are respectively the trivial group and the group of integers).

The free group of rank two is a SQ-universal group. In particular, it has subgroups that are free of every finite rank as well as a free subgroup of countable rank.

GAP implementation
The free group of rank two can be constructed using GAP with the GAP:FreeGroup command:

FreeGroup(2);

Further, the generators can also be referred to. For instance, if we use:

F := FreeGroup(2);

Then the two generators can be referred to as $$F.1$$ and $$F.2$$.