Janko group J3

Janko group J3

In mathematics, the third Janko group J3, also known as the Higman-Janko-McKay group, is a finite simple sporadic group of order 50232960. Evidence for its existence was uncovered by Zvonimir Janko, and it was shown to exist by Graham Higman and John McKay. In terms of generators a, b, c, and d its automorphism group J3:2 can be presented asa^{17} = b^8 = a^ba^{-2} = c^2 = b^cb^3 = (abc)^4 = (ac)^{17} = d^2 = [d, a] = [d, b] = (a^3b^{-3}cd)^5 = 1.A presentation for J3 in terms of (different) generators a, b, c, d isa^{19} = b^9 = a^ba^2 = c^2 = d^2 = (bc)^2 = (bd)^2 = (ac)^3 = (ad)^3 = (a^2ca^{-3}d)^3 = 1.It can also be constructed via an underlying geometry, as was done by Weiss, and has a modular representation of dimension eighteen over the finite field of nine elements, which can be expressed in terms of two generators.

J3 has a Schur multiplier of order 3, and its triple cover has a unitary 9 dimensional representation over the field with 4 elements.

J3 is one of the 6 sporadic simple groups called the pariahs, because they are not found within the Monster group.

J3 has 9 conjugacy classes of maximal subgroups:
* PSL(2,16):2, order 8160
* PSL(2,19), order 3420
* PSL(2,19), conjugate to preceding class in J3:2
* 24:(3 × A5), order 2880
* PSL(2,17), order 2448
* (3 × A6):22, order 2160
* 32+1+2:8, order 1944
* 21+4:A5, order 1920 - centralizer of involution
* 22+4:(3 × S3), order 1152

Janko predicted both J3 and J2 as simple groups having 21+4:A5 as a centralizer of an involution.

References

*Z. Janko, "Some new finite simple groups of finite order", 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 pp. 25-64 Academic Press, London, and in "The theory of finite groups" (Editied by Brauer and Sah) p. 63-64, Benjamin, 1969.MathSciNet|id=0244371
* | year=1969 | journal=Bull. London Math. Soc. | issn=0024-6093 | volume=1 | pages=89–94; correction p. 219
* Richard Weiss, "A Geometric Construction of Janko's Group J3", Math. Zeitung 179 pp 91-95 (1982)
* R. L. Griess, Jr., "The Friendly Giant", Inventiones Mathematicae 69 (1982), 1-102. p. 93: proof that J3 is a pariah.
* [http://web.mat.bham.ac.uk/atlas/v2.0/spor/J3/ Atlas of Finite Group Representations: "J"3]


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Janko group — may refer to any of the four sporadic groups named for Zvonimir Janko: * Janko group J1, of order 175560 * Hall–Janko group, of order 604800, also known as HJ and J2 * Janko group J3, of order 50232960, also known as the Higman–Janko–McKay group… …   Wikipedia

  • Janko group J1 — In mathematics, the smallest Janko group, J1, of order 175560, was first described by Zvonimir Janko (1965), in a paper which described the first new sporadic simple group to be discovered in over a century and which launched the modern theory of …   Wikipedia

  • Janko group J4 — In mathematics, the fourth Janko group J 4 is a sporadic finite simple group whose existence was suggested by Zvonimir Janko (1976), and then proven to uniquely exist by Simon Norton and others in 1980. It is the unique finite simple group of… …   Wikipedia

  • Hall–Janko group — In mathematics, the Hall Janko group HJ , is a finite simple sporadic group of order 604800. It is also called the second Janko group J 2, or the Hall Janko Wales group, since it was predicted by Janko and constructed by Hall and Wales. It is a… …   Wikipedia

  • Janko — People: *Zvonimir Janko, mathematician *Paul von Janko, inventor of the Janko keyboard *Marc Janko, Austrian football playerIn mathematics: *Janko group *Janko group J1 *Janko group J2 *Janko group J3 *Janko group J4 …   Wikipedia

  • Janko Veselinović (lawyer) — Janko Veselinović Јанко Веселиновић Born November 6, 1965(1965 11 06) Knin, Croatia Nationality Serbian …   Wikipedia

  • Conway group — Group theory Group theory …   Wikipedia

  • O'Nan group — Group theory Group theory …   Wikipedia

  • Group (mathematics) — This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines …   Wikipedia

  • Group of Lie type — In mathematics, a group of Lie type G(k) is a (not necessarily finite) group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups.… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”