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In the area of modern algebra known as group theory, the **Thompson group***Th* is a sporadic simple group of order

- 2
^{15}**·**3^{10}**·**5^{3}**·**7^{2}**·**13**·**19**·**31 - = 90745943887872000
- ≈ 9×10
^{16}.

*Th* is one of the 26 sporadic groups and was found by John G.Thompson ( 1976 ) and constructed by Smith (1976). They constructed it as the automorphism group of a certain lattice in the 248-dimensional Lie algebra of E_{8}. It does not preserve the Lie bracket of this lattice, but does preserve the Lie bracket mod 3, so is a subgroup of the Chevalley group E_{8}(3). The subgroup preserving the Lie bracket (over the integers) is a maximal subgroup of the Thompson group called the Dempwolff group (which unlike the Thompson group is a subgroup of the compact Lie group E_{8}).

The centralizer of an element of order 3 of type 3C in the Monster group is a product of the Thompson group and a group of order 3, as a result of which the Thompson group acts on a vertex operator algebra over the field with 3 elements. This vertex operator algebra contains the E_{8} Lie algebra over **F**_{3}, giving the embedding of *Th* into E_{8}(3).

The Schur multiplier and the outer automorphism group of the Thompson group are both trivial.

Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For *Th*, the relevant McKay-Thompson series is ( OEIS: A007245 ),

and *j*(*τ*) is the j-invariant.

Linton (1989) found the 16 conjugacy classes of maximal subgroups of *Th* as follows:

- 2
_{+}^{1+8}·*A* - 2
^{5}·*L*_{5}(2) This is the Dempwolff group - (3 x
*G*_{2}(3)) : 2 - (3
^{3}× 3_{+}^{1+2}) · 3_{+}^{1+2}: 2*S*_{4} - 3
^{2}· 3^{7}: 2*S*_{4} - (3 × 3
^{4}: 2 ·*A*_{6}) : 2 - 5
_{+}^{1+2}: 4*S*_{4} - 5
^{2}:*GL*_{2}(5) - 7
^{2}: (3 × 2*S*_{4}) - 31 : 15
^{3}*D*_{4}(2) : 3*U*_{3}(8) : 6*L*_{2}(19)*L*_{3}(3)*M*_{10}*S*_{5}

In the area of abstract algebra known as group theory, the **monster group** M (also known as the **Fischer–Griess monster**, or the **friendly giant**) is the largest sporadic simple group, having order

2^{46} **·** 3^{20} **·** 5^{9} **·** 7^{6} **·** 11^{2} **·** 13^{3} **·** 17 **·** 19 **·** 23 **·** 29 **·** 31 **·** 41 **·** 47 **·** 59 **·** 71

= 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

≈ 8×10^{53}.

In the area of modern algebra known as group theory, the **baby monster group***B* (or, more simply, the **baby monster**) is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Conway groups** are the three sporadic simple groups Co_{1}, Co_{2} and Co_{3} along with the related finite group Co_{0} introduced by (Conway 1968, 1969).

In the area of modern algebra known as group theory, the **Suzuki group***Suz* or *Sz* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Higman–Sims group** HS is a sporadic simple group of order

In group theory, the **Tits group**^{2}*F*_{4}(2)′, named for Jacques Tits (French: [tits]), is a finite simple group of order

In mathematics, **monstrous moonshine**, or **moonshine theory**, is the unexpected connection between the monster group *M* and modular functions, in particular, the *j* function. The term was coined by John Conway and Simon P. Norton in 1979.

In the area of modern algebra known as group theory, the **Held group***He* is a sporadic simple group of order

In the area of abstract algebra known as group theory, the **O'Nan group***O'N* or **O'Nan–Sims group** is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Rudvalis group***Ru* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Harada–Norton group***HN* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Janko group***J _{1}* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **McLaughlin group** McL is a sporadic simple group of order

In mathematical finite group theory, the **Dempwolff group** is a finite group of order 319979520 = 2^{15}·3^{2}·5·7·31, that is the unique nonsplit extension of by its natural module of order . The uniqueness of such a nonsplit extension was shown by Dempwolff (1972), and the existence by Thompson (1976), who showed using some computer calculations of Smith (1976) that the Dempwolff group is contained in the compact Lie group as the subgroup fixing a certain lattice in the Lie algebra of , and is also contained in the Thompson sporadic group (the full automorphism group of this lattice) as a maximal subgroup.

In the area of modern algebra known as group theory, the **Fischer group***Fi _{24}* or F

In the area of modern algebra known as group theory, the **Fischer group***Fi _{23}* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Fischer group***Fi _{22}* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Conway group***Co _{2}* is a sporadic simple group of order

In the area of modern algebra known as group theory, the **Conway group*** is a sporadic simple group of order*

In the area of modern algebra known as group theory, the **Conway group***Co _{1}* is a sporadic simple group of order

- Linton, Stephen A. (1989), "The maximal subgroups of the Thompson group",
*Journal of the London Mathematical Society*, Second Series,**39**(1): 79–88, doi:10.1112/jlms/s2-39.1.79, ISSN 0024-6107, MR 0989921 - Smith, P. E. (1976), "A simple subgroup of M? and E
_{8}(3)",*The Bulletin of the London Mathematical Society*,**8**(2): 161–165, doi:10.1112/blms/8.2.161, ISSN 0024-6093, MR 0409630 - Thompson, John G. (1976), "A conjugacy theorem for E
_{8}",*Journal of Algebra*,**38**(2): 525–530, doi: 10.1016/0021-8693(76)90235-0 , ISSN 0021-8693, MR 0399193

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