# 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

Type I 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.

Type II standard generators of the Thompson group Th are non-trivial elements a, b, c, d, e, s, t and u that satisfy the Havas–Soicher–Wilson presentation, see below.

A program to convert Type I to Type II generators and back again is available here in Magma format.

### 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)3y.
• 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.

### Presentation

The Soicher (or Havas–Soicher–Wilson) presentation of Th is available in Magma format here. Matrices in GAP and Magma format satisfying this presentation are given below (in the representations section). GAP matrices in separate files are also available in version 1. Note that the original words used to make these matrices seem to have been lost; the version here has been calculated anew.

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.

### Representations

The representations of Th available on Type I standard generators are:
The representation of Th available on Type II standard generators is:

### 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).

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