Rng (algebra)

The C₂ group has order 8 as shown on this circle	The C₃ (O_h) group has order 48 as shown by these spherical triangle reflection domains.

In mathematics, a hyperoctahedral group is an important type of group that can be realized as the group of symmetries of a hypercube or of a cross-polytope. It was named by Alfred Young in 1930. Groups of this type are identified by a parameter n, the dimension of the hypercube.

As a Coxeter group it is of type B_n = C_n, and as a Weyl group it is associated to the orthogonal groups in odd dimensions. As a wreath product it is $S_{2} ≀ S_{n}$ where $S_{n}$ is the symmetric group of degree n. As a permutation group, the group is the signed symmetric group of permutations π either of the set { −n, −n + 1, ..., −1, 1, 2, ..., n } or of the set { −n, −n + 1, ..., n } such that π(i) = −π(−i) for all i. As a matrix group, it can be described as the group of n×n orthogonal matrices whose entries are all integers. The representation theory of the hyperoctahedral group was described by Template:Harv according to Template:Harv.

In three dimensions, the hyperoctahedral group is known as O×S₂ where O≅S₄ is the octahedral group, and S₂ is a symmetric group (equivalently, cyclic group) of order 2. Geometric figures in three dimensions with this symmetry group are said to have octahedral symmetry. In two dimensions, the hyperoctahedral group is known as the dihedral group of order eight, describing the symmetry of a square.

By dimension

Hyperoctahedral groups can be named as BC_n, a bracket notation, or as a Coxeter group graph:

n	Symmetry group	BC_n	Coxeter notation	Coxeter group graph	Symmetry order	Structure	Related regular polytopes
2	D₄ (*4•)	BC₂	[4]	Template:CDD	2²2! = 8	$D i h_{4}$ $≅ S_{2} ≀ S_{2}$	Square, octagon
3	O_h (*432)	BC₃	[4,3]	Template:CDD	2³3! = 48	$S_{4} \times S_{2}$ $≅ S_{2} ≀ S_{3}$	Cube, octahedron
4	±¹/₆[OxO].2 ^[1] (O/V;O/V)^* ^[2]	BC₄	[4,3,3]	Template:CDD	2⁴4! = 384	$S_{2} ≀ S_{4}$	Tesseract, 16-cell, 24-cell
5		BC₅	[4,3,3,3]	Template:CDD	2⁵5! = 3840	$S_{2} ≀ S_{5}$	5-cube, 5-orthoplex
6		BC₆	[4,3⁴]	Template:CDD	2⁶6! = 46080	$S_{2} ≀ S_{6}$	6-cube, 6-orthoplex
...
n		BC_n	[4,3^n-2]	Template:CDD...Template:CDD	2ⁿn! = (2n)!!	$S_{2} ≀ S_{n}$	hypercube, orthoplex

Subgroups

There is a notable index two subgroup, corresponding to the Coxeter group D_n and the symmetries of the demihypercube. Viewed as a wreath product, there are two natural maps from the hyperoctahedral group to the cyclic group of order 2: one map coming from "multiply the signs of all the elements" (in the n copies of ${\pm 1}$ ), and one map coming from the parity of the permutation. Multiplying these together yields a third map $C_{n} \to {\pm 1}$ . The kernel of the first map is the Coxeter group $D_{n} .$ In terms of signed permutations, thought of as matrices, this third map is simply the determinant, while the first two correspond to "multiplying the non-zero entries" and "parity of the underlying (unsigned) permutation", which are not generally meaningful for matrices, but are in the case due to the coincidence with a wreath product.

The kernels of these three maps are all three index two subgroups of the hyperoctahedral group, as discussed in H₁: Abelianization below, and their intersection is the derived subgroup, of index 4 (quotient the Klein 4-group), which corresponds to the rotational symmetries of the demihypercube.

In the other direction, the center is the subgroup of scalar matrices, {±1}; geometrically, quotienting out by this corresponds to passing to the projective orthogonal group.

In dimension 2 these groups completely describe the hyperoctahedral group, which is the dihedral group Dih₄ of order 8, and is an extension 2.V (of the 4-group by a cyclic group of order 2). In general, passing to the subquotient (derived subgroup, mod center) is the symmetry group of the projective demihypercube.

The hyperoctahedral subgroup, D_n by dimension:

n	Symmetry group	D_n	Coxeter notation	Coxeter group graph	Symmetry order	Related polytopes
2	D₂ (*2•)	D₂	[ ]×[ ]	Template:CDD	4	Rectangle
3	T_d (*332)	D₃	[3,3]	Template:CDD	24	tetrahedron
4	±¹/₃[TxTemplate:Overline].2 ^[3] (T/V;T/V)_-^* ^[4]	D₄	[3^1,1,1]	Template:CDD	192	16-cell
5		D₅	[3^2,1,1]	Template:CDD	1920	5-demicube
6		D₆	[3^3,1,1]	Template:CDD	43040	6-demicube
...
n		D_n	[3^n-3,1,1]	Template:CDD...Template:CDD	2^n-1n!	demihypercube

Another notable index 2 subgroup can be called hyper-pyritohedral symmetry, by dimension:^[5] These groups have n orthogonal mirrors in n-dimensions.

n	Symmetry group	Coxeter notation	Symmetry order	Related polytopes
2	D₂ (*2•)	[4,1⁺]=[2]	4	Rectangle
3	T_h (3*2)	[4,3⁺]	24	snub octahedron
4	±¹/₃[T×T].2 ^[6] (T/V;T/V)^* ^[7]	[4,(3,3)⁺]	192	snub 24-cell
5		[4,(3,3,3)⁺]	1920
6		[4,(3,3,3,3)⁺]	43040
...
n		[4,(3^n-2)⁺]	2^n-1n!

Homology

The group homology of the hyperoctahedral group is similar to that of the symmetric group, and exhibits stabilization, in the sense of stable homotopy theory.

H₁: abelianization

The first homology group, which agrees with the abelianization, stabilizes at the Klein four-group, and is given by:

H_{1} (C_{n}, Z) = {\begin{cases} 0 & n = 0 \\ Z / 2 & n = 1 \\ Z / 2 \times Z / 2 & n \geq 2 \end{cases} .

This is easily seen directly: the $- 1$ elements are order 2 (which is non-empty for $n \geq 1$ ), and all conjugate, as are the transpositions in $S_{n}$ (which is non-empty for $n \geq 2$ ), and these are two separate classes. These elements generate the group, so the only non-trivial abelianizations are to 2-groups, and either of these classes can be sent independently to $- 1 \in {\pm 1},$ as they are two separate classes. The maps are explicitly given as "the product of the signs of all the elements" (in the n copies of ${\pm 1}$ ), and the sign of the permutation. Multiplying these together yields a third non-trivial map (the determinant of the matrix, which sends both these classes to $- 1$ ), and together with the trivial map these form the 4-group.

H₂: Schur multipliers

The second homology groups, known classically as the Schur multipliers, were computed in Template:Harv.

They are:

H_{2} (C_{n}, Z) = {\begin{cases} 0 & n = 0, 1 \\ Z / 2 & n = 2 \\ (Z / 2)^{2} & n = 3 \\ (Z / 2)^{3} & n \geq 4 \end{cases} .

Notes

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References

Template:Refbegin

Patrick du Val, Homographies, Quaternions and Rotations (1964)
John Horton Conway, On Quaternions and Octonions (2003)
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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.

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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
H.S.M. Coxeter and W. O. J. Moser. Generators and Relations for Discrete Groups 4th ed, Springer-Verlag. New York. 1980 p92, p122.

Template:Refend

Coxeter, The Beauty of Geometry: Twelve Essays (1999), Dover Publications, Template:LCCN, ISBN 0-486-40919-8

↑ Conway, 2003
↑ Du Val, 1964
↑ Conway, 2003
↑ Du Val, 1964, #42
↑ Coxeter (1999), p.121, Essay 5 Regular skew polyhedra
↑ Conway, 2003
↑ Du Val, 1964, #41

[1] Conway, 2003

[2] Du Val, 1964

[3] Conway, 2003

[4] Du Val, 1964, #42

[5] Coxeter (1999), p.121, Essay 5 Regular skew polyhedra

[6] Conway, 2003

[7] Du Val, 1964, #41

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Rng (algebra)

Contents

By dimension

Subgroups

Homology

H₁: abelianization

H₂: Schur multipliers

Notes

References

Navigation menu

Rng (algebra)

By dimension

Subgroups

Homology

H1: abelianization

H2: Schur multipliers

Notes

References

Navigation menu

Search

H₁: abelianization

H₂: Schur multipliers