Irregularity of distributions

In mathematics, the Hall–Janko group HJ, is a finite simple sporadic group of order 604800. It is also called the second Janko group J₂, or the Hall-Janko-Wales group, since it was predicted by Template:Harvs as one of two new groups with an involution centralizer of the form 2¹⁺⁴A₅ (the other is the Janko group J3) and constructed by Template:Harvs as a rank 3 permutation group on 100 points.

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. Since it is also found in the Conway group Co₁, it is therefore part of the second generation of the Happy Family.

Representations

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.

It has a modular representation of dimension six over the field of four elements; if in characteristic two we have w² + w + 1 = 0, then J₂ is generated by the two matrices

$\mathbf{A}=\left(\begin{array}{cccccc}{w}^2& w^2& 0& 0& 0& 0\\ 1& w^2& 0& 0& 0& 0\\ 1& 1& w^2& w^2& 0& 0\\ w& 1& 1& w^2& 0& 0\\ 0& w^2& w^2& w^2& 0& w\\ w^2& 1& w^2& 0& w^2& 0\end{array}\right)$

and

$\mathbf{B}=\left(\begin{array}{cccccc}w& 1& w^2& 1& w^2& w^2\\ w& 1& w& 1& 1& w\\ w& w& w^2& w^2& 1& 0\\ 0& 0& 0& 0& 1& 1\\ w^2& 1& w^2& w^2& w& w^2\\ w^2& 1& w^2& w& w^2& w\end{array}\right).$

These matrices satisfy the equations

${\mathbf{A}}^2={\mathbf{B}}^3=(\mathbf{A}\mathbf{B}{)}^7=(\mathbf{A}\mathbf{B}\mathbf{A}\mathbf{B}\mathbf{B}{)}^{12}=1.$

J₂ is thus a Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group.

The matrix representation given above constitutes an embedding into Dickson's group G₂(4). There are two conjugacy classes of HJ in G₂(4), and they are equivalent under the automorphism on the field F₄. Their intersection (the "real" subgroup) is simple of order 6048. G₂(4) is in turn isomorphic to a subgroup of the Conway group Co₁.

Maximal subgroups

J₂ has 9 conjugacy classes of maximal subgroups. Some are here described in terms of action on the Hall-Janko graph.

• U₃(3) order 6048 - one-point stabilizer, with orbits of 36 and 63

Simple, containing 36 simple subgroups of order 168 and 63 involutions, all conjugate, each moving 80 points. A given involution is found in 12 168-subgroups, thus fixes them under conjugacy. Its centralizer has structure 4.S₄, which contains 6 additional involutions.

• 3.PGL(2,9) order 2160 - has a subquotient A₆

• 2¹⁺⁴:A₅ order 1920 - centralizer of involution moving 80 points

• 2²⁺⁴:(3 × S₃) order 1152

• A₄ × A₅ order 720

Containing 2² × A₅ (order 240), centralizer of 3 involutions each moving 100 points

• A₅ × D₁₀ order 600

• PGL(2,7) order 336

• 5²:D₁₂ order 300

• A₅ order 60

Conjugacy classes

The maximum order of any element is 15. As permutations, elements act on the 100 vertices of the Hall-Janko graph.

Order	No. elements	Cycle structure and conjugacy
1 = 1	1 = 1	1 class
2 = 2	315 = 32 · 5 · 7	240, 1 class
	2520 = 23 · 32 · 5 · 7	250, 1 class
3 = 3	560 = 24 · 5 · 7	330, 1 class
	16800 = 25 · 3 · 52 · 7	332, 1 class
4 = 22	6300 = 22 · 32 · 52 · 7	26420, 1 class
5 = 5	4032 = 26 · 32 · 7	520, 2 classes, power equivalent
	24192 = 27 · 33 · 7	520, 2 classes, power equivalent
6 = 2 · 3	25200 = 24 · 32 · 52 · 7	2436612, 1 class
	50400 = 25 · 32 · 52 · 7	22616, 1 class
7 = 7	86400 = 27 · 33 · 52	714, 1 class
8 = 23	75600 = 24 · 33 · 52 · 7	2343810, 1 class
10 = 2 · 5	60480 = 26 · 33 · 5 · 7	1010, 2 classes, power equivalent
	120960 = 27 · 33 · 5 · 7	54108, 2 classes, power equivalent
12 = 22 · 3	50400 = 25 · 32 · 52 · 7	324262126, 1 class
15 = 3 · 5	80640 = 28 · 32 · 5 · 7	52156, 2 classes, power equivalent

References

• Robert L. Griess, Jr., "Twelve Sporadic Groups", Springer-Verlag, 1998.
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  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (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.)
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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Wales, David B., "The uniqueness of the simple group of order 604800 as a subgroup of SL(6,4)", Journal of Algebra 11 (1969), 455 - 460.
• Wales, David B., "Generators of the Hall-Janko group as a subgroup of G2(4)", Journal of Algebra 13 (1969), 513–516, 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park., MR0251133, ISSN 0021-8693

External links

• Atlas of Finite Group Representations: J₂