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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.
|Fitting length||not defined||There is no nontrivial nilpotent normal subgroup.|
|finitely generated group||Yes|
|slender group||No||There are free subgroups of countable rank.|
|Hopfian group||Yes||finitely generated and free implies Hopfian|
The free group of rank two can be constructed using GAP with the GAP:FreeGroup command:
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 .