Free group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

Symbol-free definition

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

• 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 generating set, or in other words, there is a generating set for the group with no nontrivial relations among the generating elements.

• 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:

• There is a generating set $S$ for $F$ such that any $g\in G$ can be uniquely expressed as a word in $S$ (that is, a product of elements from $S$ and their inverses).
• 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.

Relation with other properties

This property is a pivotal (important) member of its property space. Its variations, opposites, and other properties related to it and defined using it are often studied

• Category: Variations of freeness
• Category: Opposites of freeness

Stronger properties

Group properties stronger than the property of being free are:

• Finitely generated free group

Weaker properties

• Parafree group
• Torsion-free group
• One-relator group
• Residually nilpotent group
• Centerless group

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

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

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.