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In the area of modern algebra known as group theory, the Janko groupJ3 or the Higman-Janko-McKay groupHJM is a sporadic simple group of order
J3 is one of the 26 Sporadic groups and was predicted by Zvonimir Janko in 1969 as one of two new simple groups having 21+4:A5 as a centralizer of an involution (the other is the Janko group J2). J3 was shown to exist by GrahamHigman and John McKay ( 1969 ).
In 1982 R. L. Griess showed that J3 cannot be a subquotient of the monster group. [1] Thus it is one of the 6 sporadic groups called the pariahs.
J3 has an outer automorphism group of order 2 and a Schur multiplier of order 3, and its triple cover has a unitary 9-dimensional representation over the finite field with 4 elements. Weiss (1982) constructed it via an underlying geometry. It has a modular representation of dimension eighteen over the finite field with 9 elements. It has a complex projective representation of dimension eighteen.
J3 can be constructed by many different generators. [2] Two from the ATLAS list are 18x18 matrices over the finite field of order 9, with matrix multiplication carried out with finite field arithmetic:
and
The automorphism group J3:2 can be constructed by starting with the subgroup PSL(2,16):4 and adjoining 120 involutions, which are identified with the Sylow 17-subgroups. Note that these 120 involutions are outer elements of J3:2. One then defines the following relation:
where is the Frobenius automorphism or order 4, and is the unique 17-cycle that sends
Curtis showed, using a computer, that this relation is sufficient to define J3:2. [3]
In terms of generators a, b, c, and d its automorphism group J3:2 can be presented as
A presentation for J3 in terms of (different) generators a, b, c, d is
Finkelstein & Rudvalis (1974) found the 9 conjugacy classes of maximal subgroups of J3 as follows:
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
808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
= 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
≈ 8×1053.
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