Free group:F2

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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 $F.2$.