Janko group

In mathematics, the Janko groups J₁, J₂, J₃ and J₄ are four of the twenty-six sporadic groups; their respective orders are:

J1	$2^{3}\cdot 3\cdot 5\cdot 7\cdot 11\cdot 19$
J2	$2^{7}\cdot 3^{3}\cdot 5^{2}\cdot 7$
J3	$2^{7}\cdot 3^{5}\cdot 5\cdot 17\cdot 19$
J4	$2^{21}\cdot 3^{3}\cdot 5\cdot 7\cdot 11^{3}\cdot 23\cdot 29\cdot 31\cdot 37\cdot 43$

J₁

The smallest Janko group, J₁ of order 175560, has a presentation in terms of two generators a and b and c = abab^-1 as

$a^{2}=b^{3}=(ab)^{7}=(abc^{3})^{5}=(abc^{6}abab(ab^{-1})^{2})^{2}=1.$ It can also be expressed in terms of a permutation representation of degree 266. In fact it has 266 conjugate sugroups, simple, of order 660.

Janko found a modular representation in terms of 7 × 7 orthogonal matrices in the field of eleven elements, with generators given by

${\mathbf {Y} }=\left({\begin{matrix}0&1&0&0&0&0&0\\0&0&1&0&0&0&0\\0&0&0&1&0&0&0\\0&0&0&0&1&0&0\\0&0&0&0&0&1&0\\0&0&0&0&0&0&1\\1&0&0&0&0&0&0\end{matrix}}\right)$

and

${\mathbf {Z} }=\left({\begin{matrix}-3&2&-1&-1&-3&-1&-3\\-2&1&1&3&1&3&3\\-1&-1&-3&-1&-3&-3&2\\-1&-3&-1&-3&-3&2&-1\\-3&-1&-3&-3&2&-1&-1\\1&3&3&-2&1&1&3\\3&3&-2&1&1&3&1\end{matrix}}\right).$

J₁ was first described by Zvonimir Janko in 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 sporadic simple groups. J₁ can be characterized abstractly as the unique simple group with abelian 2-Sylow subgroups and with an involution whose centralizer is isomorphic to the direct product of the group of order two and the alternating group A_{5 of order 60, which is to say, the rotational icosahedral group. It has no outer automorphisms.}

J₁, J₃, and J₄ are among the 6 sporadic simple groups called the pariahs, because they are not found within the Monster group.

J₂

The second Janko group, of order 604800 has a presentation in terms of two generators a and b as $a^{2}=b^{3}=(ab)^{7}=(ababab^{-1}ababab^{-1}abab^{-1}ab^{-1})^{3}=1$, in terms of which it has an outer automorphism sending b to b². The group is also called the Hall-Janko group or the Hall-Janko-Wales group, since it was predicted by Janko and constructed by Hall and Wales. It is a subgroup of index two of the group of automorphisms of the Hall-Janko graph, leading to a permutation representation of degree 100. That representation has a one-point stabilizer with orbits of 36 and 63, isomorphic to the unitary group U₃(3) (order 6048).

We also may express it in terms of a modular representation of dimension six over the field of four elements; if in characteristic two we have w^{2 + w + 1 = 0, then J₂ is generated by the two matrices}

${\mathbf {A} }=\left({\begin{matrix}w^{2}&w^{2}&0&0&0&0\\w^{3}&w^{2}&0&0&0&0\\w^{3}&w^{3}&w^{2}&w^{2}&0&0\\w&w^{3}&w^{3}&w^{2}&0&0\\0&w^{2}&w^{2}&w^{2}&0&w\\w^{2}&w^{3}&w^{2}&0&w^{2}&0\end{matrix}}\right)$

and

${\mathbf {B} }=\left({\begin{matrix}w&w^{3}&w^{2}&w^{3}&w^{2}&w^{2}\\w&w^{3}&w&w^{3}&w^{3}&w\\w&w&w^{2}&w^{2}&w^{3}&0\\0&0&0&0&w^{3}&w^{3}\\w^{2}&w^{3}&w^{2}&w^{2}&w&w^{2}\\w^{2}&w^{3}&w^{2}&w&w^{2}&w\end{matrix}}\right)$

J₂ is the only one of the 4 Janko groups that is a section of the Monster group; it is thus part of what Robert Griess calls the Happy Family. It is found in the Conway group Co₁, thus part of the second generation of the Happy Family.

Griess relates [p. 123] how Marshall Hall, as editor of The Journal of Algebra, received a very short paper entitled "A simple group of order 604801." Yes, 604801 is prime.

J₃

The third Janko group, also known as the Higman-Janko-McKay group, is a finite simple sporadic group of order 50232960. Evidence for its existence was uncovered by Janko, and it was shown to exist by Higman and McKay. In terms of generators a, b, c, and d its automorphism group J₃:2 can be presented as $a^{17}=b^{8}=a^{b}a^{-2}=c^{2}=b^{c}b^{3}=(abc)^{4}=(ac)^{17}=d^{2}=[d,a]=[d,b]=(a^{3}b^{-3}cd)^{5}=1.$ A presentation for J₃ in terms of (different) generators a, b, c, d is $a^{19}=b^{9}=a^{b}a^{2}=c^{2}=d^{2}=(bc)^{2}=(bd)^{2}=(ac)^{3}=(ad)^{3}=(a^{2}ca^{-3}d)^{3}=1.$ 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.

J₄

The fourth Janko group was shown to be probable by Janko in 1976, and then proven to uniquely exist by Simon Norton in 1980. It is the unique finite simple group of order $2^{21}\cdot 3^{3}\cdot 5\cdot 7\cdot 11^{3}\cdot 23\cdot 29\cdot 31\cdot 37\cdot 43$. It has a modular representation of dimension 112 over the finite field of two elements, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. It has a presentation in terms of three generators a, b, and c as

$a^{2}=b^{3}=c^{2}=(ab)^{23}=[a,b]^{12}=[a,bab]^{5}=[c,a]=$

$(ababab^{-1})^{3}(abab^{-1}ab^{-1})^{3}=(ab(abab^{-1})^{3})^{4}=$

$[c,bab(ab^{-1})^{2}(ab)^{3}]=(bc^{bab^{-1}abab^{-1}a})^{3}=$

$((bababab)^{3}cc^{(ab)^{3}b(ab)^{6}b})^{2}=1$

External links

• PlanetMath: Janko groups
• Atlas of Finite Group Representations: J₁
• Atlas of Finite Group Representations: J₂
• Atlas of Finite Group Representations: J₃
• Atlas of Finite Group Representations: J₄

References

• Zvonimir Janko, A new finite simple group with abelian Sylow subgroups and its characterization, Journal of Algebra 32:147-186, 1966
• Daniel Gorenstein, "Finite Simple Groups", Plenum Press, 1982
• Richard Weiss, "A Geometric Construction of Janko's Group J₃", Math. Zeitung 179 pp 91-95 (1982)
• Robert L. Griess, Jr, "Twelve Sporadic Groups", Springer-Verlag, 1998.