This page is intended as place for newer texts related with "Combinatorial group theory". Older texts in this area are

• Papers on knots (1983-1984). See the page Meridional generators, ….
• The first proof of the bijectivity of the norm residue homomorphism mod 2 in degree 3 used the Reidemeister-Schreier method for subgroup presentations. See On Hilbert Satz 90 for K₃ for degree-two extensions (May 1986).
• On a cubic identity for involutive Hopf algebras (July 2022)

Generators and relations for the symmetric group

by Markus Rost (Notes, August 2026, 6 pages)

We derive the well-known Coxeter-type presentation of the symmetric group due to Moore (1896).

Moreover we mention the presentation with generators all transpositions τ_x and relations:

The elegant proof of Dickson (1899) is described.

References:

• Moore, Eliakim Hastings. Concerning the abstract groups of order k! and 1/2k! holohedrically isomorphic with the symmetric and the alternating substitution-groups on k letters. Proc. Lond. Math. Soc. 28 (1896/97), 357-366. Zbl 28.0121.03.
• Dickson, L. E. The abstract group isomorphic with the symmetric group on k letters. Proc. Lond. Math. Soc. 31 (1899), 351-353. Zbl 30.0141.03.

Full text (September 20, 2026): [pdf]

On permutation braids [announcement]

We construct the permutation braid section from the symmetric group to the braid group.

Here is a characterization of the section S in terms of the transpositions t_i=(i,i+1) (understanding S(1)=1):

Until July 2026 I was not aware of the notion of "permutation braids" and the map S. My references are

• Birman, Ko and Lee. A new approach to the word and conjugacy problems in the braid groups. Adv. Math. 139 (1998), no. 2, 322-353. MR 1654165.
• El-Rifai and Morton. Algorithms for positive braids. Quart. J. Math. Oxford Ser. (2) 45 (1994), no. 180, 479-497. MR 1315459.

I fully agree with the following remark subsequent to Lemma 2.3 in the article of El-Rifai and Morton:

The great benefit of this lemma is however that it is quite unnecessary to remember these braids as braid words; the permutation is quite enough, and makes for much greater ease in comparing such braids.

The text is in preparation since August 2026.

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