# Free group:F2

From Groupprops

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## Definition

The **free group of rank two**, also written as , is defined as the free group on a generating set of size two. is the smallest possible rank for a free non-abelian group (the free groups of rank and 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 and .