Free group

A free group is generated by a set of symbols with no relations among them beyond the group axioms. Every map from its generators to a group extends uniquely to a homomorphism.

Group: A set with an associative operation, an identity element, and an inverse for every element. Free groups are groups whose elements are built from generators while obeying these axioms.
Free reduction: The process of deleting adjacent generator-inverse pairs from a word until none remain. Multiplication concatenates words and then applies free reduction.
Group homomorphism: A function between groups that preserves the group operation. A function on free generators determines a unique homomorphism from the free group.
Tree (graph theory): A connected graph with no cycles. Free groups act naturally on trees, and their standard Cayley graphs are trees.
Group presentation: A description of a group by generators and defining relations. A free group is the presentation with chosen generators and no defining relations.
Cayley graph: A graph encoding a group's generators and multiplication through labeled edges between group elements. A free group's Cayley graph is a tree, reflecting unique reduced words.
Free product: A group combining given groups with no added relations between their elements. Free groups are free products of infinite cyclic groups, one for each basis element.
Bass–Serre theory: A theory relating group decompositions to actions on trees. It uses free groups and tree actions to study how groups split into simpler pieces.
Reduced word: A word in generators and their inverses with no adjacent symbol followed by its inverse. Reduced words give unique representatives for elements of a free group.
Word problem for groups: The decision problem of determining whether two words represent the same element of a finitely presented group. In a free group, free reduction decides whether a word represents the identity.

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