Free group:F2
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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
.