# ATLAS: Thompson group Th

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

The following information is available for Th:

### 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

#### Finding generators

To find standard generators for Th:

• Find any element x of order 2 (by taking a suitable power of any element of even order).
[The probability of success at each attempt is 18731 in 32768 (about 1 in 2).]
• Find any element of order 21 or 39. This powers up to a 3A-element, y say.
[The probability of success at each attempt is 9 in 91 (about 1 in 10).]
• Find a conjugate a of x and a conjugate b of y such that ab has order 19.
[The probability of success at each attempt is 192 in 14725 (about 1 in 77).]
• Now a and b are standard generators for Th.
This algorithm is available in computer readable format: finder for Th.

#### Checking generators

To check that elements x and y of Th are standard generators:

• Check o(x) = 2.
• Check o(y) = 3.
• Check o(xy) = 19.
• Let z = (xy)4y.
• Check o(z) = 21.
• Let w = xyy(xy)4(xyy)2(xy)2(xyy)5(xy)3.
• Check o(y(z7)w) = 2.
This algorithm is available in computer readable format: checker for Th.

### Representations

The representations of Th available are:

### Maximal subgroups

The maximal subgroups of Th are:

### Conjugacy classes

The class representatives of the 48 conjugacy classes of Th are as follows:
• 1A: identity or a2.
• 2A: a.
• 3A: b.
• 3B: (ababab2)6 or [a, bab]3.
• 3C: [a, babab]2.
• 4A: ababab2ababab2abab2.
• 4B: (abababab2ab2)3.
• 5A: [a, b]2 or (abab2)2.
• 6A: [a, babab] or (ab)3(ab2)3.
• 6B: (ab)6(ab2)3.
• 6C: (ababab2)3.
• 7A: (ab)6(ab2)6 or (abababab2)3.
• 8A: (ab)7ab2 or ((ab)3ab2ab(ab2)2)3.
• 8B: ababababab2(abab2ab2ab2)2 or (ab)9(ab2)2ab(ab2)3 or ((ab)7ab2ab(ab2)2)3.
• 9A: abab(abab2)3ab2ab2.
• 9B: ababababab2ab2abab2ab2 or ((ab)5ab2)3.
• 9C: (ababab2)2 or [a, bab].
• 10A: [a, b] or abab2.
• 12A/B: abababab2ababab2ab2.
• 12C: (abababab2ab2)2ab2.
• 12D: abababab2ab2.
• 13A: (ab)9(ab2)3 or ab(abababab2)2.
• 14A: (ab)4(ab2)3.
• 15A/B: (ab)6ab2abab2ab2.
• 18A: (ab)10(ab2)4 or abababab2abab2ab2abab2 or (ab)5(ababab2)2.
• 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 or ab(abababab2)2abab2.
• 28A: (ab)6ab2.
• 30A/B: (ab)5ab2abab2ab2.
• 31A/B: (ab)5(ab2)2ab(ab2)4 or ab(ababab2)3ab2.
• 36A: (ab)8(ab2)2.
• 36B/C: ab(abababab2)2ab2ab2.
• 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.

An amalgam:
• Words to generate an amalgam of N(3B) and N(3B2) over a common subgroup of index 4 (in both groups) are given here (Magma format). The copy of N(3B) is the same as that given the maximal subgroups section; the copy of N(3B2) is a different representative.

A couple of non-maximal subgroups with few prime divisors.
• 51+2:42:2, of order 4000 and index 3 in a conjugate of Max9, with generators a^((ab)5(abb)14), ((abababababb)5)^((abb)9(ab)6).
The class distribution of this subgroup in Th is (1A1, 2A175, 4B1000, 5A124, 8B1000, 10A700, 20A1000).
• 51+2:D8, of order 1000, with generators a^((ab)5(abb)2), ((abababababb)5)^((abb)17(ab)8).
This subgroup has no normal 52, so does not lie in a conjugate of Max10.
The class distribution of this subgroup in Th is (1A1, 2A125, 4B50, 5A124, 10A500, 20A200).

The information below is not currently available at this site. Try following this link to the version 1 site at the Birmingham ‘mirror’ of this Atlas.
Matrices in dimension 248 over GF(2) satisfying the Soicher (or Havas–Soicher–Wilson) presentation of Th are available in GAP format by following the links given below.

a , b , c , d , e , s , t , u .

These have been specially imported from version 1, and do not satisfy our version 2 conventions. Also a and b do not refer to standard generators as they do elsewhere on this page. The relevant reference is:

G.Havas, L.H.Soicher and R.A.Wilson. A presentation for the Thompson sporadic simple group. Groups and Computation III (Columbus, Ohio, 1999), 193–200, Ohio State Univ. Math. Res. Inst. Publ. 8, de Gruyter, Berlin, 2001.

Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old Th page - ATLAS version 1.
Anonymous ftp access is also available. See here for details.

Version 2.0 created on 14th April 1999.
Last updated 03.06.10 by JNB.
Information checked to Level 1 on 11.06.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.