By Wilhelm Magnus

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**Additional resources for Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations**

**Example text**

Such that S, T, ... are derivable from P, Q, R, ... , and conversely. Show that the equivalence class group (a, b, c, ... ; P, Q, R, ... ) is the same as the equivalence class group (a, b, c, ... ; S, T, ... ). 4. Show that the equivalence class group P, Q, R, ... ) (a, b, c, •.. ; is the same as the equivalence class group of each of the following: (a, b, c, ... ; p-l, Q, R, ... • ; RPQ, Q, R, ... • ; WPW-I, Q, R, •. ); P', Q, R, ... ). (a, b, c, ... ; where P' is a cyclic permutation of idi+1' .

Show that the equivalence class group (a, b, c, ... ; P, Q, R, ... ) is the same as the equivalence class group (a, b, c, ... ; S, T, ... ). 4. Show that the equivalence class group P, Q, R, ... ) (a, b, c, •.. ; is the same as the equivalence class group of each of the following: (a, b, c, ... ; p-l, Q, R, ... • ; RPQ, Q, R, ... • ; WPW-I, Q, R, •. ); P', Q, R, ... ). (a, b, c, ... ; where P' is a cyclic permutation of idi+1' . ·infd2· . ·ii-I· 5. Suppose G Then show = (a, b; p. , if P =id2' .

LbPt ... a·rbiJr , and can define the identity element. (h) Conclude that one set of canonical forms for (a, b) consists of the empty word, together with the words of the form where oc l or {Jr may be zero, but otherwise all OCi and fJi are non-zero integers. This solves the word problem for the free group (a, b). (i) Conclude also that (a, b) is a presentation for the group generated by and 32 SEC. 3 10. 'Show that canonical forms for where p is a positive prime, are where "'lor Pr may be zero, but otherwise 0 < "'i < p, 0 < Pj < p.