Standard Coxeter Presentations

On this page, we give definitions of the standard Coxeter presentations. We also intend to list many finite examples of them-hopefully all of them.

The groups Gp,q,r

< a, b, c | a2 = b2 = c2 = (ab)p = (ac)2 = (bc)q = (abc)r = 1 >

Remark: The isomorphism-type of the group defined by this presentation is independent of the ordering of p, q and r. It has a total length of 10 + 2p + 2q + 3r. Note that Gp,q,2r contains (2, p, q; r) [see below] to index 2. Here, the (2, p, q; r) is on generators ca and ab.

The groups (p, q, r)

< a, b, c | ap = bq = cr = abc = 1 >
or
< a, b | ap = bq = (ab)r = 1 >

Remark: The group obtained is independent of the order of p, q and r. The second presentation is obtained from the first by letting c = (ab)-1. (Obviously (ab)r = 1 is equivalent to (ab)-r = 1.) The former presentation has length 3 + p + q + r and the latter presentation has length p + q + 2r.

The groups (p, q, r; s)

< a, b, c | ap = bq = cr = abc = (cba)s = 1 >
or
< a, b | ap = bq = (ab)r = [a, b]s = 1 >

Remark: The group obtained is independent of the order of p, q and r. The second presentation is obtained from the first by letting c = (ab)-1. (Obviously (ab)r = 1 is equivalent to (ab)-r = 1. Also, cba = [b, a] = [a, b]-1.) The former presentation has length 3 + p + q + r + 3s and the latter presentation has length p + q + 2r + 4s.

The groups (p, q | r, s)

< a, b | ap = bq = (ab)r = (ab-1)s = 1 >

Remark: The group obtained is independent of swapping p with q and r with s. This presentation has length p + q + 2r + 2s.

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Last updated 23rd August, 1997
John N. Bray