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Group theory and generalizations Definition

Tietze transformations

Builds on Presentation of a groupFree groupNormal closureGroup isomorphism

Definition

Let ⟨S∣R⟩\langle S \mid R \rangle be a group presentation, with RR a subset of the free group F(S)F(S). Let N(R)N(R) be the normal closure of RR in F(S)F(S). The Tietze transformations are the following four operations.

  1. Add a relation: replace RR by R∪{r}R \cup \set{r}, where r∈N(R)r \in N(R).
  2. Remove a relation: replace RR by R∖{r}R \setminus \set{r}, where r∈Rr \in R lies in the normal closure of R∖{r}R \setminus \set{r}.
  3. Add a generator: for a new symbol x∉Sx \notin S and a word w∈F(S)w \in F(S), replace ⟨S∣R⟩\langle S \mid R \rangle by ⟨S∪{x}∣R∪{xw−1}⟩\langle S \cup \set{x} \mid R \cup \set{x w^{-1}} \rangle.
  4. Remove a generator: suppose x∈Sx \in S and RR contains a relation xw−1x w^{-1}, with ww a word in S∖{x}S \setminus \set{x}. Delete xx and this relation, and replace xx by ww in all other relations.

Remarks

Each transformation yields a presentation of an isomorphic group. Moves 1 and 2 keep N(R)N(R) unchanged, and in moves 3 and 4 the added or removed letter equals an existing element. Tietze's theorem: two finite presentations define isomorphic groups if and only if one is obtained from the other by a finite sequence of Tietze transformations. Example: ⟨a,b∣aba−1b−2, b⟩\langle a, b \mid a b a^{-1} b^{-2},\ b \rangle becomes ⟨a∣aa−1⟩\langle a \mid a a^{-1} \rangle by removing bb with w=1w = 1, and then ⟨a∣ ⟩\langle a \mid \ \rangle by removing the trivial relation; so it presents ℤ\Z. Non-example: replacing ⟨a∣a2⟩\langle a \mid a^2 \rangle by ⟨a∣a4⟩\langle a \mid a^4 \rangle is not a Tietze transformation, since a2a^2 is not in the normal closure of a4a^4; indeed the groups ℤ/2\Z/2 and ℤ/4\Z/4 differ. Tietze transformations do not give an algorithm for the group isomorphism problem, which is undecidable for finite presentations.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Free group, concept, no evidence levelFree groupconceptGroup isomorphism, concept, no evidence levelGroup isomorphismconceptNormal closure, concept, no evidence levelNormal closureconceptPresentation of a group, concept, no evidence levelPresentation of a gro…conceptTietze transformations, definition, no evidence levelTietze transformationsdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Free group concept
  2. Group isomorphism concept
  3. Normal closure concept
  4. Presentation of a group concept
  5. Tietze transformations definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Tietze_transformations
About this page
TagTDCGM4WR88
URIm2://TDCGM4WR88
Kinddefinition
WritingW0 stub
MSC20F05
DefinesTietze transformations
UsesPresentation of a group, Free group, Normal closure, Group isomorphism
Changed2 November 2026

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