Quaternion Group
| 1 | | 1 | | | | | | | | | | | | | 1 | | | | | | | | | 1 | | | | | | | | | 1 | | | | | 1 | | | | | 1 | | | | | | | | | 1 | | | | | | | | | 1 | | | | | |
The cycle graph of the quaternion group is illustrated above.
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Weisstein, Eric W. "Quaternion Group." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/QuaternionGroup.html
Subject classifications
quaternion group
or, equivalently, defined by the multiplication table
Quaternions were known to Gauss in 1819 or 1820, but he did not publicize this discovery, and quaternions weren’t rediscovered until 1843, with Hamilton.
| Title | quaternion group |
|---|---|
| Canonical name | QuaternionGroup |
| Date of creation | 2013-03-22 12:35:35 |
| Last modified on | 2013-03-22 12:35:35 |
| Owner | mathcam (2727) |
| Last modified by | mathcam (2727) |
| Numerical id | 12 |
| Author | mathcam (2727) |
| Entry type | Definition |
| Classification | msc 20A99 |
| Synonym | quaternionic group |
| Related topic | Quaternions |
| Defines | quaternion group |
Quaternion group
In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication. It is often denoted by Q or Q8, and is given by the group presentation
\( \mathrm{Q} = \langle -1,i,j,k \mid (-1)^2 = 1, \;i^2 = j^2 = k^2 = ijk = -1 \rangle, \,\! \)
where 1 is the identity element and −1 commutes with the other elements of the group.
ompared to dihedral group
The Q8 group has the same order as the dihedral group D4, but a different structure, as shown by their Cayley and cycle graphs:
The dihedral group D 4 arises in the split-quaternions in the same way that Q 8 lies in the quaternions.
Cayley table
The Cayley table (multiplication table) for Q is given by:[1]
| Q×Q | 1 | −1 | − | − | − | |||
|---|---|---|---|---|---|---|---|---|
| 1 | 1 | −1 | − | − | − | |||
| −1 | −1 | 1 | − | − | − | |||
| − | −1 | 1 | − | − | ||||
| − | − | 1 | −1 | − | − | |||
| − | − | −1 | 1 | − | ||||
| − | − | − | 1 | −1 | − | |||
| − | − | − | −1 | 1 | ||||
| − | − | − | − | 1 | −1 |
The multiplication of the six imaginary units {±i, ±j, ±k} works like the cross product of unit vectors in three-dimensional Euclidean space.
\( \begin{alignat}{2} ij & = k, & ji & = -k, \\ jk & = i, & kj & = -i, \\ ki & = j, & ik & = -j. \end{alignat} \)
The quaternion group has the unusual property of being Hamiltonian: every subgroup of Q is a normal subgroup, but the group is non-abelian.[2] Every Hamiltonian group contains a copy of Q.[3]
In abstract algebra, one can construct a real four-dimensional vector space with basis {1, i, j, k} and turn it into an associative algebra by using the above multiplication table and distributivity. The result is a skew field called the quaternions. Note that this is not quite the same as the group algebra on Q (which would be eight-dimensional). Conversely, one can start with the quaternions and define the quaternion group as the multiplicative subgroup consisting of the eight elements {1, −1, i, −i, j, −j, k, −k}. The complex four-dimensional vector space on the same basis is called the algebra of biquaternions.
Note that i, j, and k all have order four in Q and any two of them generate the entire group. Another presentation of Q[4] demonstrating this is:
\( \langle x,y \mid x^4 = 1, x^2 = y^2, y^{-1}xy = x^{-1}\rangle.\,\! \)
One may take, for instance, i = x, j = y and k = x y.
The center and the commutator subgroup of Q is the subgroup {±1}. The factor group Q/{±1} is isomorphic to the Klein four-group V. The inner automorphism group of Q is isomorphic to Q modulo its center, and is therefore also isomorphic to the Klein four-group. The full automorphism group of Q is isomorphic to S4, the symmetric group on four letters. The outer automorphism group of Q is then S4/V which is isomorphic to S3.
Matrix representations
The quaternion group can be represented as a subgroup of the general linear group GL2(C). A representation
\( \mathrm{Q} = \{\pm 1, \pm i, \pm j, \pm k\} \to \mathrm{GL}_{2}(\mathbf{C}) \)
is given by
\( 1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)
\( i \mapsto \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} \)
\( j \mapsto \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \)
\( k \mapsto \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix} \)
Since all of the above matrices have unit determinant, this is a representation of Q in the special linear group SL 2 ( C ). The standard identities for quaternion multiplication can be verified using the usual laws of matrix multiplication in GL 2 ( C ). [5]
There is also an important action of Q on the eight nonzero elements of the 2-dimensional vector space over the finite field F 3 . A representation
\( \mathrm{Q} = \{\pm 1, \pm i, \pm j, \pm k\} \to \mathrm{GL}(2,3) \)
\( i \mapsto \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \)
\( j \mapsto \begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix} \)
\( k \mapsto \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \)
where {−1,0,1} are the three elements of F3. Since all of the above matrices have unit determinant over F3, this is a representation of Q in the special linear group SL(2, 3). Indeed, the group SL(2, 3) has order 24, and Q is a normal subgroup of SL(2, 3) of index 3.
Galois group
As Richard Dean showed in 1981, the quaternion group can be presented as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field, over Q, of the polynomial
\( x^8 - 72 x^6 + 180 x^4 - 144 x^2 + 36. \)
The development uses the fundamental theorem of Galois theory in specifying four intermediate fields between Q and T and their Galois groups, as well as two theorems on cyclic extension of degree four over a field.[6]
Generalized quaternion group
A group is called a generalized quaternion group or dicyclic group if it has a presentation[4]
\( \langle x,y \mid x^{2n} = y^4 = 1, x^n = y^2, y^{-1}xy = x^{-1}\rangle.\,\! \)
for some integer n ≥ 2. This group is denoted Q4n and has order 4n.[7] Coxeter labels these dicyclic groups <2,2,n>, being a special case of the binary polyhedral group <l,m,n> and related to the polyhedral groups (p,q,r), and dihedral group (2,2,n). The usual quaternion group corresponds to the case n = 2. The generalized quaternion group can be realized as the subgroup of GL2(C) generated by
\( \left(\begin{array}{cc} \omega_n & 0 \\ 0 & \overline{\omega}_n \end{array} \right) \mbox{ and } \left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right) \)
where ωn = eiπ/n.[4] It can also be realized as the subgroup of unit quaternions generated by[8] x = eiπ/n and y = j.
The generalized quaternion groups have the property that every abelian subgroup is cyclic. [9] It can be shown that a finite p -group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above. [10] Another characterization is that a finite p -group in which there is a unique subgroup of order p is either cyclic or generalized quaternion (of order a power of 2). [11] In particular, for a finite field F with odd characteristic, the 2-Sylow subgroup of SL 2 ( F ) is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group, (Gorenstein 1980, p. 42). Letting p r be the size of F , where p is prime, the size of the 2-Sylow subgroup of SL 2 ( F ) is 2 n , where n = ord 2 ( p 2 − 1) + ord 2 ( r ).
The Brauer–Suzuki theorem shows that groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.
binary tetrahedral group Clifford algebra dicyclic group Hurwitz integral quaternion List of small groups 16-cell
See also a table from Wolfram Alpha See Hall (1999), p. 190 See Kurosh (1979), p. 67 Johnson 1980, pp. 44–45 Artin 1991 Dean, Richard (1981). "A Rational Polynomial whose Group is the Quaternions". The American Mathematical Monthly 88 (1): 42–45. JSTOR 2320711. Some authors (e.g., Rotman 1995, pp. 87, 351) refer to this group as the dicyclic group, reserving the name generalized quaternion group to the case where n is a power of 2. Brown 1982, p. 98 Brown 1982, p. 101, exercise 1 Cartan & Eilenberg 1999, Theorem 11.6, p. 262
Brown 1982, Theorem 4.3, p. 99
Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-13-004763-2 Brown, Kenneth S. (1982), Cohomology of groups (3 ed.), Springer-Verlag, ISBN 978-0-387-90688-1 Cartan, Henri; Eilenberg, Samuel (1999), Homological Algebra, Princeton University Press, ISBN 978-0-691-04991-5 Coxeter, H. S. M. and Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9. Dean, Richard A. (1981) "A rational polynomial whose group is the quaternions", American Mathematical Monthly 88:42–5. Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 81b:20002 Johnson, David L. (1980), Topics in the theory of group presentations, Cambridge University Press, ISBN 978-0-521-23108-4, MR 0695161 Rotman, Joseph J. (1995), An introduction to the theory of groups (4 ed.), Springer-Verlag, ISBN 978-0-387-94285-8 P.R. Girard (1984) "The quaternion group and modern physics", European Journal of Physics 5:25–32. Hall, Marshall (1999), The theory of groups (2 ed.), AMS Bookstore, ISBN 0-8218-1967-4 Kurosh, Alexander G. (1979), Theory of Groups, AMS Bookstore, ISBN 0-8284-0107-1
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A hypercomplex number , geometrically realizable in four-dimensional space. The system of quaternions was put forward in 1843 by W.R. Hamilton (1805–1865). Quaternions were historically the first example of a hypercomplex system, arising from attempts to find a generalization of complex numbers. Complex numbers are depicted geometrically by points in the plane and operations on them correspond to the simplest geometric transformations of the plane. It is not possible to "organize" a number system similar to the field of real or complex numbers from the points of a space of three or more dimensions. However, if one drops the requirement of commutativity of multiplication, then it is possible to construct a number system from the points of $ 4 $- dimensional space. (In 3, 5 or higher-dimensional space it is not even possible to do this.)
The quaternions form a $ 4 $- dimensional algebra over the field of real numbers with basis $ 1 , i , j , k $( "basic units" ) and the following multiplication table of the "basic units" :
<tbody> </tbody>
Every quaternion can be written in the form $$ X = x _ {0} \cdot 1 + x _ {1} \cdot i + x _ {2} \cdot j + x _ {3} \cdot k $$ or (since 1 plays the role of ordinary identity and in writing a quaternion it can be omitted) in the form $$ X = x _ {0} + x _ {1} i + x _ {2} j + x _ {3} k . $$ One distinguishes the scalar part $ x _ {0} $ of the quaternion and its vector part $$ V = x _ {1} i + x _ {2} j + x _ {3} k , $$ so that $ X = x _ {0} + V $. If $ x _ {0} = 0 $, then the quaternion $ V $ is called a vector and can be identified with an ordinary $ 3 $- dimensional vector, since multiplication in the algebra of quaternions of two such vectors $ V _ {1} $ and $ V _ {2} $ is related to the scalar and vector products $ ( V _ {1} , V _ {2} ) $( cf. Inner product ) and $ [ V _ {1} , V _ {2} ] $( cf. Vector product ) of the vectors $ V _ {1} $ and $ V _ {2} $ in $ 3 $- dimensional space by the formula $$ V _ {1} V _ {2} = \ - ( V _ {1} , V _ {2} ) + [ V _ {1} , V _ {2} ] . $$ This shows the close relationship between quaternions and vector calculus . Historically, the latter arose from the theory of quaternions. Corresponding to each quaternion $ X = x _ {0} + V $ is the conjugate quaternion $ \overline{X}\; = x _ {0} - V $, and $$ X \cdot \overline{X}\; = \overline{X}\; \cdot X = \ x _ {0} ^ {2} + x _ {1} ^ {2} + x _ {2} ^ {2} + x _ {3} ^ {2} . $$ This real number is called the norm of the quaternion $ X $ and is denoted by $ N ( X) $. This norm satisfies the relation $$ N ( XY ) = N ( X) N ( Y) . $$ Any rotation of $ 3 $- dimensional space about the origin can be defined by means of a quaternion $ P $ with norm 1. The rotation corresponding to $ P $ takes the vector $ X = x _ {1} i + x _ {2} j + x _ {3} k $ to the vector $ Y = y _ {1} i + y _ {2} j + y _ {3} k = P X P ^ {-} 1 $. The algebra of quaternions is the unique associative non-commutative finite-dimensional normed algebra over the field of real numbers with an identity. The algebra of quaternions is a skew-field , that is, division is defined in it, and the quaternion inverse to a quaternion $ X $ is $ \overline{X}\; / N ( X) $. The skew-field of quaternions is the unique finite-dimensional real associative non-commutative algebra without divisors of zero (see also Frobenius theorem ; Cayley–Dickson algebra ).
Let $ \zeta $ be the element $ ( 1+ i+ j+ k)/2 $ in the algebra of quaternions. The Hurwitz ring of integral quaternions is the ring $$ H = \{ {m _ {0} \zeta + m _ {1} i + m _ {2} j + m _ {3} k } : { m _ {0} , m _ {1} , m _ {2} , m _ {3} \in \mathbf Z } \} . $$ The Hurwitz ring is a non-commutative ring in which an analogue of the Euclidean division property (cf. Euclidean algorithm ) holds: For any $ a, b \in H $ with $ b \neq 0 $ there exist elements $ q, r \in H $ such that $$ a = qb + r $$ $$ N( r) < N( b) . $$ (This property does not hold for the subring $ \{ {n _ {0} + n _ {1} i + n _ {2} j + n _ {3} k } : {n _ {0} , n _ {1} , n _ {2} , n _ {3} \in \mathbf Z } \} $.) It follows that every left ideal is left principal, and this in turn can be used to give a proof of the Lagrange four-square theorem, to the effect that every positive integer can be written as a sum of four squares of integers. The Lagrange identity, which plays an important role in the proof of this result, $$ ( a _ {0} ^ {2} + a _ {1} ^ {2} + a _ {2} ^ {2} + a _ {3} ^ {2} ) ( b _ {0} ^ {2} + b _ {1} ^ {2} + b _ {2} ^ {2} + b _ {3} ^ {2} ) = $$ $$ = \ ( a _ {0} b _ {0} - a _ {1} b _ {1} - a _ {2} b _ {2} - a _ {3} b _ {3} ) ^ {2} + $$ $$ + ( a _ {0} b _ {1} + a _ {1} b _ {0} + a _ {2} b _ {3} - a _ {3} b _ {2} ) ^ {2} + $$ $$ + ( a _ {0} b _ {2} - a _ {1} b _ {3} + a _ {2} b _ {0} + a _ {3} b _ {1} ) ^ {2} + $$ $$ + ( a _ {0} b _ {3} + a _ {1} b _ {2} - a _ {2} b _ {1} + a _ {3} b _ {0} ) ^ {2} $$ for real numbers $ a _ {0,\ } a _ {1} , a _ {2} , a _ {3} , b _ {0} , b _ {1} , b _ {2} , b _ {3} $, is equivalent to the multiplicativity of the norm $ N( XY) = N( X) N( Y) $, where $ X $, $ Y $ are the quaternions $ X = a _ {0} + a _ {1} i + a _ {2} j + a _ {3} k $, $ Y = b _ {0} + b _ {1} i + b _ {2} j + b _ {3} k $. Writing $ X $ as $ ( a _ {0} + a _ {1} i) + ( a _ {2} + a _ {3} i ) j $ and putting $ \alpha = a _ {0} + a _ {1} i $, $ \beta = a _ {2} + a _ {3} i $, one obtains $ X = \alpha + \beta j $. It is easily proved that the algebra of quaternions is isomorphic to the algebra of complex $ ( 2 \times 2) $- matrices $$ \left ( \begin{array}{cc} \alpha &\beta \\ \overline \beta \; &\overline \alpha \; \\ \end{array} \right ) , $$ with $ \overline \alpha \; , \overline \beta \; $ the complex conjugates of $ \alpha , \beta \in \mathbf C $. When one wishes to retain the multiplicativity of the norm, $ N( XY) = N( X) N( Y) $, there is only one possible generalization of the quaternions (over the reals): the octaves or octonions, which have 8 instead of 4 components (Hurwitz's theorem, 1898; cf. Cayley numbers ). The centre of the skew-field of quaternions is the field of real numbers. Later the notion of hypercomplex system has been generalized in a theory of skew-fields over arbitrary fields, e.g. the theory of the Brauer group of a commutative field. In this connection, a generalized quaternion algebra is a $ 4 $- dimensional algebra over a field $ F $ generated by $ 1, x, y , xy $ with multiplication table $ x ^ {2} = a $, $ y ^ {2} = b $, $ yx = - xy $, where $ a, b $ are non-zero elements of $ F $. (The quaternions are the case $ a = b = - 1 $, and $ F $ the field of real numbers.)
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Price includes VAT (Russian Federation) Instant access to the full article PDF. Rent this article via DeepDyve Institutional subscriptions Literature CitedN. Bourbaki, Algebra [Russian translation], Fizmatgiz, Moscow (1962). Google Scholar I. M. Gel'fand and M. I. Graev, "Representations of groups of quaternions over an unconnected locally compact continuous field," Dokl. Akad. Nauk SSSR, 177 , No. 1 (1967). G. Mackey, "Infinite dimensional group representations," Matematika, 6 , No. 6, 57–103 (1962). I. M. Gel'fand and M. I. Graev, "Representations of groups of second-order matrices with elements from a locally compact field," Usp. Mate. Nauk, 18 , No. 4, 29–99 (1962). Also see I. M. Gel'fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions [in Russian], Nauka, Moscow (1966). I. M. Gel'fand, R. A. Minlos, and Z. Ya. Shapiro, Representations of Rotation Groups and Lorentz Groups [in Russian], Fizmatgiz, Moscow (1958). Download references You can also search for this author in PubMed Google Scholar Additional informationLomonosov Moscow State University and Institute of Applied Mathematics of the Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 2, No. 1, pp. 20–35, January–March, 1968. Rights and permissionsReprints and permissions About this articleGel'fand, I.M., Graev, M.I. Representations of quaternion groups over locally compact and functional fields. Funct Anal Its Appl 2 , 19–33 (1968). https://doi.org/10.1007/BF01075357 Download citation Received : 16 October 1967 Issue Date : January 1968 DOI : https://doi.org/10.1007/BF01075357 Share this articleAnyone you share the following link with will be able to read this content: Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative
Stack Exchange NetworkStack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Presentations of the dihedral and quaternion groups Note: I understand that he is using typical notation for the quaternion group but it is easy to see that any pair of quaternions which satisfy those relations will generate the group as well.
2 Answers 2In addition to Gerry Myerson's comments, I would like to offer another perspective: what I personally would understand by reading this is that Lang is defining a presentation of a group and defining it to be $D_8$ or $Q_8$ . What I mean by this that, in the case of the dihedral group, we ignore any visualizations with the square - just take a generating set $\{\sigma, \tau\}$ and form the group $\langle \sigma,\tau\mid \sigma^4=e,\tau^2=e,\tau\sigma\tau^{-1}=\sigma^3\rangle$ , which is a definition of some group, which we call "the group of symmetries of the square", or $D_8$ . The same applies for the quaternions. Conversely, once we define $D_8,Q_8$ like this, we cannot take any old elements $\sigma',\tau'\in D_8$ that satisfy the relations and claim that they generate the group - what if $\sigma'=\tau'=e$ ? We would need to additionally require that $\sigma',\tau'$ generate a group of order 8. 
It might be well to recall the way that presentations of groups work. The presentation $\langle S|R\rangle $ refers to the group which is the quotient of the free group on the generators, the elements of $S $ , by the normal subgroup, or the so-called normal closure, of the subgroup generated by the relations $R $ . You must log in to answer this question.Not the answer you're looking for browse other questions tagged soft-question finite-groups quaternions group-presentation dihedral-groups ..
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In group theory, the quaternion group Q 8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset of the quaternions under multiplication. It is given by the group presentation. where e is the identity element and e commutes with the other elements of the group.
The group presentation of the quaternion group is given by: Dic2 = a, b: a4 = e,b2 =a2, aba = b Dic 2 = a, b: a 4 = e, b 2 = a 2, a b a = b .
The quaternion group is one of the two non-Abelian groups of the five total finite groups of order 8. It is formed by the quaternions , , , and , denoted or . The multiplication table for is illustrated above, where rows and columns are given in the order , , , , 1, , , , as in the table above. The cycle graph of the quaternion group is ...
That is, any quaternion hmay be written uniquely as h= z+ jw (z;w2C); (0.1d) the multiplication rules are determined by (0.1c). There is an algebra anti-automorphism of the quaternions given by a+ bi+ cj+ dk= a bi cj dk; h 1h 2 = h 2h 1: (0.1e) A right quaternionic vector space V is automatically a complex vector
Quaternions are multiplied according to the foil method. Just expand every-thing out, and then use the rules above to simplify the expression so that it looks like a quaternion. For instance (4+j)(i+k) = 4i+4k +ji+jk = 4i+4k −k +i = 5i+3k. Quaternions are often denoted by single variables, like complex numbers. For instance q = a+bi+cj +dj. 1
The quaternion group from a semi-direct product The group Q 8 is built out of its subgroups hiiand hjiwith the condition i2 = j2 = 1 and the conjugacy relation jij 1 = i= i 1. More generally, for odd awe have jaij a= i= i 1, and for even awe have jaij a= i. We can combine these into one formula:
generalized quaternion group. The groups given by the presentation. Q4n = a,b:an = b2,a2n =1,b−1ab = a−1 . are the generalized quaternion groups. Generally one insists that n>1 as the properties of generalized quaternions become more uniform at this stage. However if n =1 then one observes a = b2 so Q4n ≅Z4.
The quaternion group is a Hamilton group, and the minimal Hamilton group in the sense that any non-Abelian Hamilton group contains a subgroup isomorphic to the quaternion group. The intersection of all non-trivial subgroups of the quaternion group (and also of any generalized quaternion group) is a non-trivial subgroup.
The quaternion group, or quaternionic group, is a noncommutative group with eight elements. It is traditionally denoted by Q Q (not to be confused with Q ℚ) or by Q8 Q 8. This group is defined by the presentation. or, equivalently, defined by the multiplication table. where we have put each product xy x y into row x x and column y y .
Byanalogy with the group ofquaternion matrices over an unconnected locally compact field K, rep-resentations of a group G of arbitrary rank will be constructed in this section. Description of Representations of Even Rank n 2m. = Let us introduce the subgroup A=mC SU (2, C {t)) of matrices of the form.
The quaternion group is the smallest dicyclic group. It is the smallest example of the so-called Hamiltonian groups, which are groups every subgroup of which is a normal group. Every Hamiltonian group contains a copy of . It can be also given by the generating relations or , or as a group generated by two elements and , both of order 4 subject ...
N.8 = a * b^2 * a. N.9 = a * b^-1 * a * b. The second point is that the first relation a4 = e a 4 = e in your presentation is actually redundant. The quaternion group is defined by the presentation with two generators a, b a, b and just the two relations a2 = b2 a 2 = b 2 and abab−1 = e a b a b − 1 = e. It is a nice exercise to prove that!
Yet, simple presentations together with physical applications seem to be scarce. In the following, I shall show that several of the major covariance groups of physics can easily be related to a finite group, namely, the abstract quaternion ... The abstract quaternion group (denoted Q) which was discovered by William Rowan Hamilton (Hankins 1980 ...
We present the quaternion group. This is an important example of a non-abelian group of small order.http://www.michael-penn.nethttp://www.randolphcollege.edu...
Quaternion group. In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication. It is often denoted by Q or Q8, and is given by the group presentation. [Math Processing Error]
Proposition 1.6. Assume Gis a group, p: G!G=[G;G] is a canonical projection. For every one-dimensional representation : G=[G;G] !F we de ne the one-dimensional representation : G!F by = p. The map 7! is a bijection between the set of one-dimensional representation of the group G=[G;G] and the set of one-dimensional representation of the group G.
Quaternion. A hypercomplex number, geometrically realizable in four-dimensional space. The system of quaternions was put forward in 1843 by W.R. Hamilton (1805-1865). Quaternions were historically the first example of a hypercomplex system, arising from attempts to find a generalization of complex numbers.
3. Consider the group G = x, y ∣ xn =y2,x2n = 1,y−1xy =x−1 . As noted by @AgenorAndrade, this group has order at least 4n, as it has your group as a homomorphic image. To show that it as at most 4n elements, just note that every element of G can be written as xiyj, for 0 ≤ i <2n, and j ∈ {0, 1}. In fact, in an arbitrary product of x ...
I. M. Gel'fand and M. I. Graev, "Representations of groups of quaternions over an unconnected locally compact continuous field," Dokl. Akad. Nauk SSSR,177, No. 1 (1967). ... Quaternion Group; Functional Field; Use our pre-submission checklist. Avoid common mistakes on your manuscript. Advertisement. Search. Search by keyword or author. Search.
What I mean by this that, in the case of the dihedral group, we ignore any visualizations with the square - just take a generating set {σ, τ} {σ, τ} and form the group σ, τ ∣ σ4 = e,τ2 = e, τστ−1 =σ3 σ, τ ∣ σ 4 = e, τ 2 = e, τ σ τ − 1 = σ 3 , which is a definition of some group, which we call "the group of symmetries ...