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.

Arithmetic functions

Function	Value	Explanation
order	infinite (countable)
exponent	infinite (countable)
Fitting length	not defined	There is no nontrivial nilpotent normal subgroup.

Group properties

Property	Satisfied	Explanation	Comment
cyclic group	No
abelian group	No
nilpotent group	No
solvable group	No
free group	Yes
hypocentral group	Yes
hypoabelian group	Yes
imperfect group	Yes
finitely generated group	Yes
slender group	No	There are free subgroups of countable rank.
centerless group	Yes
Hopfian group	Yes	finitely generated and free implies Hopfian
co-Hopfian group	No

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 $F.2$.