# ATLAS: Thompson group Th

Order = 90745943887872000 = 215.310.53.72.13.19.31.
Mult = 1.
Out = 1.

### Standard generators

Standard generators of the Thompson group Th are a and b where a has order 2, b is in class 3A and ab has order 19.

### Black box algorithms

To find standard generators for Th:
• Find any element x of order 2.
• Find any element of order 21 or 39. This powers up to a 3A-element, y, say.
• Find a conjugate a of x and a conjugate b of y such that ab has order 19.
• a and b are now standard generators for Th.

### Representations

The representations of Th available are:
• a and b as 248 × 248 matrices over GF(2).
• a and b as 248 × 248 matrices over GF(3).
• a and b as 3875 × 3875 matrices over GF(3).
• a and b as 248 × 248 matrices over GF(5).
• a and b as 248 × 248 matrices over GF(7).
• a and b as 248 × 248 matrices over GF(13).
• a and b as 248 × 248 matrices over GF(19).
• a and b as 248 × 248 matrices over GF(31).
• a and b as 248 × 248 matrices over Q.

### Maximal subgroups

The maximal subgroups of Th are

### Conjugacy classes

(Some of) The class representatives of the 48 conjugacy classes of Th are as follows:
• 1A: identity.
• 2A: a.
• 3A: b.
• 3B: (ababab2)6.
• 3C: .
• 4A: .
• 4B: .
• 5A: [a, b]2 or (abab2)2.
• 6A: .
• 6B: .
• 6C: (ababab2)3.
• 7A: (ab)6(ab2)6.
• 8A: .
• 8B: .
• 9A: .
• 9B: .
• 9C: (ababab2)2.
• 10A: [a, b] or abab2.
• 12A/B: .
• 12C: .
• 12D: .
• 13A: (ab)9(ab2)3 or ab(abababab2)2.
• 14A: (ab)4(ab2)3.
• 15A/B: .
• 18A: (ab)10(ab2)4 or abababab2abab2ab2abab2.
• 18B: ababab2.
• 19A: ab.
• 20A: (ab)4ab2.
• 21A: abababab2.
• 24A/B: (ab)3ab2ab(ab2)2.
• 24C/D: (ab)7ab2ab(ab2)2.
• 27A: (ab)5ab2.
• 27B/C: (ab)7ab2abab(ab2)3.
• 28A: (ab)6ab2.
• 30A/B: (ab)5ab2abab2ab2.
• 31A/B: (ab)5(ab2)2ab(ab2)4.
• 36A: (ab)8(ab2)2.
• 36B/C: .
• 39A/B: (ab)8ab2ab(ab2)2.
An element cannot be obtained as a power of an element of greater order just if it is in class 18B or has order at least 19.

A set of generators for the maximal cyclic subgroups can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Problems of algebraic conjugacy are not dealt with.

Work in progress: GAP-format matrices for elements a, b, c, d, e, s, t, u, for Leonard Soicher.
Return to main ATLAS page.

Last updated 7th January 2000,
R.A.Wilson, R.A.Parker and J.N.Bray