Free group

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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VIEW RELATED: Group property implications | Group property non-implications | Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions |Group property dissatisfactions<random> @@@
RANDOM GROUP PROPERTY: <random>Simple group: A nontrivial group having no proper nontrivial normal subgroup.@@@Noetherian group: A group where every subgroup is finitely generated.@@@Complete group: A group for which the natural homomorphism to its automorphism group is an isomorphism, i.e., a centerless group for which every automorphism is inner.@@@SQ-universal group: A group for which every finitely generated group can be realized as a subquotient.@@@T-group: A group where any subnormal subgroup is normal, i.e., where normality is transitive. In general, normality is not transitive.@@@Locally finite group: A group in which every finitely generated subgroup is finite.@@@Group satisfying normalizer condition: A group with no proper self-normalizing subgroup.</random></random>

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Definition

Symbol-free definition

A group is said to be free if it satisfies the following equivalent conditions:

1. 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.
2. 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

Note that any generating set which satisfies the first property also satisfies the second, and vice versa. Such a generating set is said to be a freely generating set.

Definition with symbols

A group $F$ is said to be free if it satisfies the following equivalent conditions:

1. There is a generating set $S$ for $F$ such that any $g\in F$ 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).
2. There is a subset $S$ of $F$ such that given any set-theoretic map $f$ from $S$ to a group $G$, there is a unique group homomorphism $h_{f}$ from $F$ to $G$ whose restriction to $S$ is $f$

Note that any generating set $S$ which satisfies the first property also satisfies the second, and vice versa. Such a generating set is said to be a freely generating set.

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				|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
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			|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

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		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.