Free Group

• Let be any, not necessarily finite, set of elements . We refer to as an alphabet, each of as letters, any symbol of the form , is a syllable and a finite string of syllables concatenated is a word. The empty word is which has no syllables.

• There are two elementary contractions that can be done in a word. A reduced word is one where no more elementary contractions can be performed.

  • Replace all occurrences of to .
  • Replace all occurrences of with .

• Let be an alphabet. The group called the free group generated by is defined as follows.

  Let be the set of reduced all reduced words from . The group operation is defined by concatenation. That is for the product is the reduced form of the word obtained by concatenation.

• If is a group, with a set of generators and if is isomorphic to under a map such that then is free on , each are free generators of . A group is free if it is free on some nonempty set . That is .

• (Fraleigh 39.7) If a group is free on and also on , then and have the same number of elements. Any two sets of free generators of a free group have the same cardinality.

• The rank of a group free on is .

• (Fraleigh 39.9) Two free groups are isomorphic if and only if they have the same rank.

• Nielsen–Schreier theorem (Fraleigh 39.10) A nontrivial proper subgroup of a free group is free.

• (Fraleigh 39.12) Let be generated by and let be any group. If for are any elements in , not necessarily distinct, then there is at most one homomorphism such that . If is free on (i.e., ) , then there is exactly one such homomorphism

  A homomorphism of a group is completely determined if we know its value on each element of a generating set.

  • Proof: For any , the generator sets imply that:

    Now

    So indeed is uniquely determined by the value on each element of a generating set.

    If then define by

    Since only has reduced words, no two formal products in are equal (i.e., is uniquely determined) and clearly is a homomorphism.

• (Fraleigh 39.13) Every group is a homomorphic image of a free group .

Links §

• A First Course in Abstract Algebra 7th Edition by Fraleigh