Conceptual clustering

In group theory, a branch of mathematics, extra special groups are analogues of the Heisenberg group over finite fields whose size is a prime. For each prime p and positive integer n there are exactly two (up to isomorphism) extra special groups of order p¹⁺²ⁿ. Extra special groups often occur in centralizers of involutions. The ordinary character theory of extra special groups is well understood.

Definition

Recall that a finite group is called a p-group if its order is a power of a prime p.

A p-group G is called extra special if its center Z is cyclic of order p, and the quotient G/Z is a non-trivial elementary abelian p-group.

Extra special groups of order p¹⁺²ⁿ are often denoted by the symbol p¹⁺²ⁿ. For example, 2¹⁺²⁴ stands for an extra special group of order 2²⁵.

Classification

Every extra special p-group has order p¹⁺²ⁿ for some positive integer n, and conversely for each such number there are exactly two extra special groups up to isomorphism. A central product of two extra special p-groups is extra special, and every extra special group can be written as a central product of extra special groups of order p³. This reduces the classification of extra special groups to that of extra special groups of order p³. The classification is often presented differently in the two cases p odd and p = 2, but a uniform presentation is also possible.

p odd

There are two extra special groups of order p³, which for p odd are given by

• The group of triangular 3x3 matrices over the field with p elements, with 1's on the diagonal. This group has exponent p for p odd (but exponent 4 if p=2).
• The semidirect product of a cyclic group of order p² by a cyclic group of order p acting non-trivially on it. This group has exponent p².

If n is a positive integer there are two extra special groups of order p¹⁺²ⁿ, which for p odd are given by

• The central product of n extra special groups of order p³, all of exponent p. This extra special group also has exponent p.
• The central product of n extra special groups of order p³, at least one of exponent p². This extra special group has exponent p².

The two extra special groups of order p¹⁺²ⁿ are most easily distinguished by the fact that one has all elements of order at most p and the other has elements of order p².

p = 2

There are two extra special groups of order 8 = 2³, which are given by

• The dihedral group D₈ of order 8, which can also be given by either of the two constructions in the section above for p = 2 (for p odd they given different groups, but for p = 2 they give the same group). This group has 2 elements of order 4.
• The quaternion group Q₈ of order 8, which has 6 elements of order 4.

If n is a positive integer there are two extra special groups of order 2¹⁺²ⁿ, which are given by

• The central product of n extra special groups of order 8, an odd number of which are quaternion groups. The corresponding quadratic form (see below) has Arf invariant 1.
• The central product of n extra special groups of order 8, an even number of which are quaternion groups. The corresponding quadratic form (see below) has Arf invariant 0.

The two extra special groups G of order 2¹⁺²ⁿ are most easily distinguished as follows. If Z is the center, then G/Z is a vector space over the field with 2 elements. It has a quadratic form q, where q is 1 if the lift of an element has order 4 in G, and 0 otherwise. Then the Arf invariant of this quadratic form can be used to distinguish the two extra special groups. Equivalently, one can distinguish the groups by counting the number of elements of order 4.

All p

A uniform presentation of the extra special groups of order p¹⁺²ⁿ can be given as follows. Define the two groups:

• $M(p)=\langle a,b,c:a^{p}=b^{p}=1,c^{p}=1,ba=abc,ca=ac,cb=bc\rangle$
• $N(p)=\langle a,b,c:a^{p}=b^{p}=c,c^{p}=1,ba=abc,ca=ac,cb=bc\rangle$

M(p) and N(p) are non-isomorphic extra special groups of order p³ with center of order p generated by c. The two non-isomorphic extra special groups of order p¹⁺²ⁿ are the central products of either n copies of M(p) or n−1 copies of M(p) and 1 copy of N(p). This is a special case of a classification of p-groups with cyclic centers and simple derived subgroups given in Template:Harv.

Character theory

If G is an extra special group of order p¹⁺²ⁿ, then its irreducible complex representations are given as follows:

• There are exactly p²ⁿ irreducible representations of dimension 1. The center Z acts trivially, and the representations just correspond to the representations of the abelian group G/Z.
• There are exactly p−1 irreducible representations of dimension pⁿ. There is one of these for each non-trivial character χ of the center, on which the center acts as multiplication by χ. The character values are given by pⁿχ on Z, and 0 for elements not in Z.

• If a nonabelian p-group G has less than p²-p nonlinear irreducible characters of minimal degree, it is extraspecial.

Examples

It is quite common for the centralizer of an involution in a finite simple group to contain a normal extra special subgroup. For example, the centralizer of an involution of type 2B in the monster group has structure 2¹⁺²⁴.Co_{1, which means that it has a normal extra special subgroup of order 2¹⁺²⁴, and the quotient is one of the Conway groups.}

Generalizations

Groups whose center, derived subgroup, and Frattini subgroup are all equal are called special groups. Infinite special groups whose derived subgroup has order p are also called extra special groups. The classification of countably infinite extra special groups is very similar to the finite case, Template:Harv, but for larger cardinalities even basic properties of the groups depend on delicate issues of set theory, some of which are exposed in Template:Harv. The nilpotent groups whose center is cyclic and derived subgroup has order p and whose conjugacy classes are at most countably infinite are classified in Template:Harv. Finite groups whose derived subgroup has order p are classified in Template:Harv.

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
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  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
  
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  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
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  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
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  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
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  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