Free group

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RANDOM GROUP PROPERTY: SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.

This term is related to: geometric group theory
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This term is related to: combinatorial group theory
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Definition

No.	Shorthand	A group is termed free if ...	A group $G$ is termed free if ...
1	free product of copies of integers	it is the internal free product of (possibly infinitely many) groups, each of which is isomorphic to the group of integers. Hence, it is also isomorphic to the external free product of copies of the group of integers.	there are subgroups $G_i, i \in I$ of $G$ such that $G$ is the internal free product of the $G_i$s and each $G_i$ is isomorphic to $\mathbb{Z}$, the group of integers.
2	freely generating set, in terms of unique reduced words	there is a generating set for the group such that every element of the group can uniquely be expressed as a reduced word in terms of the elements of the generating set (and their inverses), with the multiplication being by concatenation of words.	there is a generating set $S$ for $G$ such that any $g \in G$ can be uniquely expressed as a reduced word in $S$ (that is, a product of elements from $S$ and their inverses, with no letter occurring adjacent to its inverse).
3	freely generating set, in terms of universal property	there is a subset of the group, such that any set-theoretic map from that subset to any target group, lifts uniquely to a group homomorphism from the whole group to the target group.	there is a subset $S$ of $G$ such that given any set-theoretic map $f$ from $S$ to a group $H$, there is a unique group homomorphism $h_f$ from $G$ to $H$ whose restriction to $S$ is $f$.
4	projective object in the category of groups	any surjective homomorphism from another group to it splits, i.e., there is an injective homomorphism backward such that the one-way composite is the identity.	for any surjective homomorphism $\varphi:H \to G$, there is an injective homomorphism $\alpha:G \to H$ such that $\varphi(\alpha(g)) = g$ for all $g \in G$.

Note that the notions of freely generating set described in formulations (2) and (3) of the definition are equivalent.

Equivalence of definitions

Further information: Equivalence of definitions of free group

Formalisms

Category-theoretic formulation

We can consider the free group functor: the functor that associates to any set, the group generated freely by that set. This is a functor because any map of sets gives rise to a map of the corresponding free groups.

The free group functor can be defined as the left adjoint to the forgetful functor from groups to sets. In other words, if $\mathcal{U}$ denotes the forgetful functor from groups to sets (that sends a group to its underlying set) and $\mathcal{F}$ denotes the free group functor, then for any set $A$ and group $G$, there is a natural isomorphism of sets:

$\operatorname{Hom}(\mathcal{F}(A),G) = \operatorname{Hom}(A,\mathcal{U}(G))$

where the left side is the set of group homomorphisms and the right set is the set of set homomorphisms (i.e., all the set-theoretic maps).

Examples

• The free group on the empty set is the trivial group (this isn't typically considered a free group).
• The free group on a set of size one is isomorphic to the group of integers $\mathbb{Z}$, i.e., it is infinite cyclic. it is the only Abelian nontrivial free group).
• The free group on a set of size two is an important free group. It is non-Abelian, finitely generated, and is SQ-universal: every finitely generated group is a subquotient of this group

Relation with other properties

Stronger properties

Group properties stronger than the property of being free are:

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Finitely generated free group	both free and a finitely generated group, or equivalently, free on a finite generating set			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
parafree group	residually nilpotent and has the same lower central series as a free group, with the isomorphism realized by a group homomorphism from the free group.			|FULL LIST, MORE INFO
reduced free group	free in some subvariety of the variety of groups; quotient of a free group by a verbal subgroup			|FULL LIST, MORE INFO
torsion-free group (also called aperiodic group)	no non-identity element has finite order			|FULL LIST, MORE INFO
group in which every abelian subgroup is cyclic				|FULL LIST, MORE INFO
one-relator group				|FULL LIST, MORE INFO
residually nilpotent group	its lower central series members intersect at the identity			|FULL LIST, MORE INFO
centerless group				|FULL LIST, MORE INFO
residually solvable group	its derived series members intersect trivially			Residually nilpotent group|FULL LIST, MORE INFO
group satisfying Tits alternative				|FULL LIST, MORE INFO
group that is the characteristic closure of a singleton subset				|FULL LIST, MORE INFO
residually finite group	every non-identity element is outside a normal subgroup of finite index			|FULL LIST, MORE INFO

Incomparable properties

Property	Meaning	Proof of one non-implication	Proof of other non-implication	Properties stronger than both	Properties weaker than both
Free abelian group	free object in variety of abelian groups	free groups of rank 2 or more are not abelian, hence definitely not free abelian.	free abelian groups that are not cyclic cannot be free because if the group is free and abelian, it must have rank at most one and hence be cyclic	Only two such groups: trivial group and group of integers	Group in which every fully invariant subgroup is verbal, Reduced free group|FULL LIST, MORE INFO
Complete group	centerless, every automorphism is inner; equivalently, injective in the category of groups			Only the trivial group has both properties.	|FULL LIST, MORE INFO

Facts

The cardinalities of any two freely generating sets of the same free group are equal. This result actually follows from the fact that the corresponding result is true for free Abelian groups.

For full proof, refer: Free groups satisfy IBN

This cardinality is termed the rank of the free group. It is further clear that any two free groups of the same rank are isomorphic.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
Subgroup-closed group property	Yes	freeness is subgroup-closed	If $G$ is a free group, and $H$ is a subgroup of $G$, then $H$ is also a free group.
Quotient-closed group property	No	every group is a quotient of a free group	A quotient of a free group need not be free. In fact, for any group $G$, there is a free group $F$ having $G$ as a quotient.
Finite-direct product-closed group property	No		We can have free groups $F_1$ and $F_2$ such that $F_1 \times F_2$ is not free (in fact, if both are nontrivial, the direct product is definitely not free).
Free product-closed group property	Yes	free product of free groups is free	If $G_i,i \in I$are free groups, so is their free product.

More information on these metaproperties: [SHOW MORE]

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties

Every subgroup of a free group is free. This result is fairly nontrivial. In general, however, the number of generators of the subgroup could be bigger or smaller than the number of generators of the whole group.

For full proof, refer: Freeness is subgroup-closed

Quotient-closedness

The property of being free is far from quotient-closed -- in fact, the quotient-closure of the property of being free is the property of being any group, viz any group can be expressed as a quotient of a free group by a normal subgroup.

In fact, if we take an arbitrary group $G$, and a generating set $S$ for $G$, we can express $G$ as the quotient of the free group $F(S)$ by the normal subgroup comprising all words in elements of $S$ that reduce to the identity element in $G$.

This is the idea behind a presentation of a group, for instance.

Direct product-closedness

A direct product of free groups is not free. This follows from the fact that we get commutation relation between the elements in the two direct factors, but a free group is centerless.

Free product-closedness

A free product of free groups is free. Here, the freely generating set of the free product is the union of the freely generating sets of the free factors.