nLab
Tietze transformation

Tietze transformations

Idea

Tietze transformations are a formalisation of the informal substitution methods that are natural when working with group presentations.

The four transformations

Let G=X:RG= \langle X: R\rangle be a group presentation, where the ‘specified isomorphism to GG’ is unspecified!

The following transformations do not change the group GG:

T1: Adding a superfluous relation

X:R\langle X: R\rangle becomes X:R \langle X: R^'\rangle, where R =R{r}R^' = R \cup \{r\} and rN(R)r\in N(R) the normal closure of the relations in the free group on XX, i.e., rr is a consequence of RR;

T2: Removing a superfluous relation

X:R\langle X: R\rangle becomes X:R \langle X: R^'\rangle where R =R{r}R^' = R - \{r\}, and rr is a consequence of R R^';

T3: Adding a superfluous generator

X:R\langle X: R\rangle becomes X :R \langle X^': R^'\rangle, where X =X{g}X^' = X\cup \{ g\}, gg being a new symbol not in XX, and R =R{wg 1}R^' = R\cup\{wg^{-1}\}, where ww is a word in the other generators, that is ww is in the image of the inclusion of F(X)F(X) into F(X )F(X^');

T4: Removing a superfluous generator

X:R\langle X: R\rangle becomes X :R \langle X^': R^'\rangle, where X =X{g}X^' = X - \{ g\}, and R =R{wg 1}R^' = R-\{wg^{-1}\} with wF(X )w\in F(X^') and wg 1Rwg^{-1}\in R and no other members of RR\prime involve gg.

Tietze’s theorem

Theorem

Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze transformations.

References

Tietze’s original paper is

  • H. Tietze, Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatsschr. Math. Phys., 19 (1908) 1 –118.

See also

  • W. Magnus and B. Chandler, The history of combinatorial group theory, Springer (1982).

category: group theory

Revised on January 30, 2012 17:42:03 by Tim Porter (95.147.237.107)