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| [[Image:GroupDiagramQ8.svg|right|thumb|[[Cycle graph (group)|Cycle diagram]] of Q. Each color specifies a series of powers of any element connected to the identity element (1). For example, the cycle in red reflects the fact that ''i'' <sup>2</sup> = −1, ''i'' <sup>3</sup> = −''i'' and ''i'' <sup>4</sup> = 1. The red cycle also reflects the fact that (−''i'' )<sup>2</sup> = −1, (−''i'' )<sup>3</sup> = ''i'' and (−''i'' )<sup>4</sup> = 1.]]
| | There are numerous proven methods to treat hemorrhoids which are powerful for lengthy expression hemorrhoid relief. Because, while hemorrhoids, like numerous other illnesses have a genetic component - when a mom or daddy had them you're more likely to receive them - they furthermore are influenced by lifestyle. Some of the factors which contribute to the occurrence of hemorrhoids are chronic constipation, sitting for extended periods of time, and a sofa potato lifestyle. So, hemorrhoids will be cured or at least place into significant remission by utilizing some good sense life-style techniques. I will focus on the easiest of these techniques to implement.<br><br>Another sort of [http://hemorrhoidtreatmentfix.com/bleeding-hemorrhoids stop bleeding hemorrhoids] is the pill that is to be swallowed. It assists to regulate blood flows plus stress inside the body system. This helps to tighten vein tissues plus this delivers a relaxation for the hemorrhoid region. This nonetheless has certain side effects that may not augur well for some individuals. Pharmacists may actually not prescribe this option as a result of the negative effects.<br><br>Yes, hemorrhoids are painful, irritating and embarrassing. Folks wish a rapid fix for which. This is where the lotions come inside, plus they do function. Except they never stop the hemorrhoids from coming back. Would it not create more sense to find out what exactly is causing them and deal with which when and for all?<br><br>Lunch plus Dinner. Gradually add real fruits plus veggies to the meals, plus substitute entire grains for white flour and pasta for an additional fiber punch.<br><br>Well, he HAD heard about a hemorrhoids house remedy or 2, he mentioned, yet couldn't absolutely remember any details regarding them. I told him I needed time to consider the upcoming step and got from there plus into the bright sunlight because quick because I may. Surgery for hemorrhoids surely wasn't my first choice.<br><br>The operation procedures commonly involve scalpel, this is commonly done by cutting away the swelling and closing up the wounds. With this procedure you're required to stay inside the hospital following the procedure so which you will be provided some painkillers. Normally you'll feel the serious pain when the anesthesia wears off.<br><br>Piles may change each aspect of your lifetime, from how commonly we take a seat and curl about how much food you eat plus for many; it could feel like we can not do anything to create your hemorrhoids disappear. While many persons may have the periodic flare up, a lot of individuals have piles that have progressed into a long-term situation. As an example, many fresh mothers suffer from hemorrhoids from labour. The problem is the fact that based on the intensity, you can possibly end up suffering with this problem again, plus again plus again. |
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| In [[group theory]], the '''quaternion group''' is a [[nonabelian group|non-abelian]] [[group (mathematics)|group]] of [[Group order|order]] eight, isomorphic to a certain eight-element subset of the [[quaternion]]s under multiplication. It is often denoted by Q or Q<sub>8</sub>, and is given by the [[presentation of a group|group presentation]]
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| :<math>\mathrm{Q} = \langle -1,i,j,k \mid (-1)^2 = 1, \;i^2 = j^2 = k^2 = ijk = -1 \rangle, \,\!</math>
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| where 1 is the identity element and −1 [[commutativity|commutes]] with the other elements of the group.
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| == Cayley graph ==
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| The Q<sub>8</sub> group has the same order as the [[Dihedral group]], [[Examples_of_groups#The_symmetry_group_of_a_square_-_dihedral_group_of_order_8|D<sub>4</sub>]], but a different structure, as shown by their Cayley graphs:
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| {| class=wikitable width=480
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| |+ [[Cayley graph]]
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| |- align=center valign=top
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| |[[Image:Cayley graph Q8.svg|240px]]<BR>Q<sub>8</sub><BR>The red arrows represent multiplication on the right by ''i'', and the green arrows represent multiplication on the right by ''j''.
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| |[[File:Cayley Graph of Dihedral Group D4.svg|240px]]<BR>D<sub>4</sub><BR>[[Dihedral group]]
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| |}
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| == Cayley table ==
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| The [[Cayley table]] (multiplication table) for Q is given by:<ref>See also [http://www.wolframalpha.com/input/?i=Quaternion+group a table] from [[Wolfram Alpha]]</ref>
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| {|class="wikitable" style="margin: auto; text-align: center;"
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| ! !! 1 !! −1 !! i !! −i !! j !! −j !! k !! −k
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| |-
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| ! 1
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| | 1 || −1 || i || −i || j || −j || k || −k
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| |-
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| ! −1
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| | −1 || 1 || −i || i || −j || j || −k || k
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| |-
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| ! i
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| | i || −i || −1 || 1 || k || −k || −j || j
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| |-
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| ! −i
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| | −i || i || 1 || −1 || −k || k || j || −j
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| |-
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| ! j
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| | j || −j || −k || k || −1 || 1 || i || −i
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| |-
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| ! −j
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| | −j || j || k || −k || 1 || −1 || −i || i
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| |-
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| ! k
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| | k || −k || j || −j || −i || i || −1 || 1
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| |-
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| ! −k
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| | −k || k || −j || j || i || −i || 1 || −1
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| |}
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| The multiplication of pairs of elements from the subset {±''i'', ±''j'', ±''k''} works like the [[cross product]] of [[unit vectors]] in three-dimensional [[Euclidean space]].
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| :<math>\begin{alignat}{2}
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| ij & = k, & ji & = -k, \\
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| jk & = i, & kj & = -i, \\
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| ki & = j, & ik & = -j.
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| \end{alignat}</math>
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| == Properties ==
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| The quaternion group has the unusual property of being [[Hamiltonian group|Hamiltonian]]: every [[subgroup]] of Q is a [[normal subgroup]], but the group is non-abelian.<ref>See Hall (1999), [http://books.google.com/books?id=oyxnWF9ssI8C&pg=PA190 p. 190]</ref> Every Hamiltonian group contains a copy of Q.<ref>See Kurosh (1979), [http://books.google.com/books?id=rp9c0nyjkbgC&pg=PA67 p. 67]</ref>
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| 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 [[quaternion]]s. 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 [[biquaternion]]s.
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| Note that ''i'', ''j'', and ''k'' all have [[order (group theory)|order]] four in Q and any two of them generate the entire group. Another [[presentation of a group|presentation]] of Q<ref name="Johnson44-45">{{harvnb|Johnson|1980|loc=pp. 44–45}}</ref> demonstrating this is:
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| :<math>\langle x,y \mid x^4 = 1, x^2 = y^2, y^{-1}xy = x^{-1}\rangle.\,\!</math>
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| One may take, for instance, ''i'' = ''x'', ''j'' = ''y'' and ''k'' = ''x y''.
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| The [[center of a group|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 S<sub>4</sub>, the [[symmetric group]] on four letters. The [[outer automorphism group]] of Q is then S<sub>4</sub>/V which is isomorphic to S<sub>3</sub>.
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| == Matrix representations ==
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| [[File:Quaternion group; Cayley table; subgroup of SL(2,C).svg|thumb|'''Q. g.''' as a subgroup of [[Special linear group|SL]](2,[[Complex number|'''C''']])]]
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| The quaternion group can be [[group representation|represented]] as a subgroup of the [[general linear group]] GL<sub>2</sub>('''C'''). A representation
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| :<math>\mathrm{Q} = \{\pm 1, \pm i, \pm j, \pm k\} \to \mathrm{GL}_{2}(\mathbf{C})</math>
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| is given by
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| :<math>1 \mapsto \begin{pmatrix}
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| 1 & 0 \\
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| 0 & 1
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| \end{pmatrix}</math>
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| :<math>i \mapsto \begin{pmatrix}
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| i & 0 \\
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| 0 & -i
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| \end{pmatrix}</math>
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| :<math>j \mapsto \begin{pmatrix}
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| 0 & 1 \\
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| -1 & 0
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| \end{pmatrix}</math>
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| :<math>k \mapsto \begin{pmatrix}
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| 0 & i \\
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| i & 0
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| \end{pmatrix}</math>
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| Since all of the above matrices have unit determinant, this is a representation of Q in the [[special linear group]] SL<sub>2</sub>('''C'''). The standard identities for quaternion multiplication can be verified using the usual laws of matrix multiplication in GL<sub>2</sub>('''C''').<ref>{{harvnb|Artin|1991}}</ref>
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| [[File:Quaternion group; Cayley table; subgroup of SL(2,3).svg|thumb|'''Q. g.''' as a subgroup of [[:File:SL(2,3); Cayley table.svg|SL(2,3)]] ]]
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| There is also an important action of Q on the eight nonzero elements of the 2-dimensional vector space over the [[finite field]] '''F'''<sub>3</sub>. A representation
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| :<math>\mathrm{Q} = \{\pm 1, \pm i, \pm j, \pm k\} \to \mathrm{GL}(2,3)</math>
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| is given by
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| :<math>1 \mapsto \begin{pmatrix}
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| 1 & 0 \\
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| 0 & 1
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| \end{pmatrix}</math>
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| :<math>i \mapsto \begin{pmatrix}
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| 1 & 1 \\
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| 1 & -1
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| \end{pmatrix}</math>
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| :<math>j \mapsto \begin{pmatrix}
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| -1 & 1 \\
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| 1 & 1
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| \end{pmatrix}</math>
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| :<math>k \mapsto \begin{pmatrix}
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| 0 & -1 \\
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| 1 & 0
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| \end{pmatrix}</math>
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| where {−1,0,1} are the three elements of '''F'''<sub>3</sub>. Since all of the above matrices have unit determinant over '''F'''<sub>3</sub>, 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 of a subgroup|index]] 3.
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| ==Galois group==
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| As [[Richard Dean (mathematician)|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 number]]s and ''T'' is the [[splitting field]], over Q, of the polynomial
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| :<math>x^8 - 72 x^6 + 180 x^4 - 144 x^2 + 36</math>.
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| 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.<ref>{{cite journal
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| | last = Dean | first = Richard
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| | authorlink = Richard Dean (mathematician)
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| | year = 1981
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| | title = A Rational Polynomial whose Group is the Quaternions
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| | journal = [[American Mathematical Monthly|The American Mathematical Monthly]]
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| | volume = 88 | issue = 1 | pages = 42–45
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| | jstor = 2320711
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| }}</ref>
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| ==Generalized quaternion group==
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| A group is called a '''generalized quaternion group''' or [[dicyclic group]] if it has a [[presentation of a group|presentation]]<ref name="Johnson44-45"/>
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| :<math>\langle x,y \mid x^{2n} = y^4 = 1, x^n = y^2, y^{-1}xy = x^{-1}\rangle.\,\!</math>
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| for some integer {{nowrap|''n'' ≥ 2}}. This group is denoted Q<sub>4''n''</sub> and has order 4''n''.<ref>Some authors (e.g., {{harvnb|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.</ref> [[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 group]]s (p,q,r), and dihedral group (2,2,n). The usual quaternion group corresponds to the case {{nowrap|1=''n'' = 2}}. The generalized quaternion group can be realized as the subgroup of GL<sub>2</sub>('''C''') generated by
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| :<math>\left(\begin{array}{cc}
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| \omega_n & 0 \\
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| 0 & \overline{\omega}_n
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| \end{array}
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| \right)
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| \mbox{ and }
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| \left(\begin{array}{cc}
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| 0 & -1 \\
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| 1 & 0
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| \end{array}
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| \right)
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| </math>
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| where ω<sub>''n''</sub> = e<sup>iπ/''n''</sup>.<ref name="Johnson44-45"/> It can also be realized as the subgroup of unit quaternions generated by<ref>{{harvnb|Brown|1982}}, p. 98</ref> {{nowrap|1=''x'' = e<sup>iπ/''n''</sup>}} and {{nowrap|1=''y'' = j}}.
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| The generalized quaternion groups have the property that every [[abelian group|abelian]] subgroup is cyclic.<ref>{{harvnb|Brown|1982}}, p. 101, exercise 1</ref> It can be shown that a finite [[p-group|''p''-group]] with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above.<ref>{{harvnb|Cartan|Eilenberg|1999}}, Theorem 11.6, p. 262</ref> 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).<ref>{{harvnb|Brown|1982}}, Theorem 4.3, p. 99</ref> In particular, for a finite field ''F'' with odd characteristic, the 2-Sylow subgroup of SL<sub>2</sub>(''F'') is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group, {{harv|Gorenstein|1980|p=42}}. Letting ''p<sup>r</sup>'' be the size of ''F'', where ''p'' is prime, the size of the 2-Sylow subgroup of SL<sub>2</sub>(''F'') is 2<sup>''n''</sup>, where {{nowrap|1=''n'' = ord<sub>2</sub>(''p''<sup>2</sup> − 1) + ord<sub>2</sub>(''r'')}}.
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| The [[Brauer–Suzuki theorem]] shows that groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.
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| ==See also==
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| *[[binary tetrahedral group]]
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| *[[Clifford algebra]]
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| *[[dicyclic group]]
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| *[[Hurwitz integral quaternion]]
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| *[[List of small groups]]
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| *[[16-cell]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{citation
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| |author-link=Michael Artin
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| |last=Artin
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| |first=Michael
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| |title=Algebra
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| |publisher=Prentice Hall
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| |year=1991
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| |isbn=978-0-13-004763-2
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| }}
| |
| *{{citation
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| |last=Brown
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| |first=Kenneth S.|authorlink=Kenneth Brown (mathematician)
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| |title=Cohomology of groups
| |
| |publisher=Springer-Verlag
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| |year=1982
| |
| |edition=3
| |
| |isbn=978-0-387-90688-1
| |
| }}
| |
| *{{citation
| |
| |author-link=Henri Cartan
| |
| |last=Cartan
| |
| |first=Henri
| |
| |last2=Eilenberg
| |
| |first2=Samuel
| |
| |author2-link=Samuel Eilenberg
| |
| |title=Homological Algebra
| |
| |publisher=Princeton University Press
| |
| |year=1999
| |
| |isbn=978-0-691-04991-5
| |
| }}
| |
| *{{cite book | author=[[H. S. M. Coxeter|Coxeter, H. S. M.]] and Moser, W. O. J. | title=Generators and Relations for Discrete Groups | location=New York | publisher=Springer-Verlag | year=1980 | isbn=0-387-09212-9}}
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| * Dean, Richard A. (1981) "A rational polynomial whose group is the quaternions", [[American Mathematical Monthly]] 88:42–5.
| |
| *{{Citation | last1=Gorenstein | first1=D. | author1-link=Daniel Gorenstein | title=Finite Groups | publisher=Chelsea | location=New York | isbn=978-0-8284-0301-6 | mr=81b:20002 | year=1980}}
| |
| *{{citation
| |
| |last=Johnson
| |
| |first=David L.
| |
| |title=Topics in the theory of group presentations
| |
| |publisher=[[Cambridge University Press]]
| |
| |isbn=978-0-521-23108-4
| |
| |year=1980
| |
| |mr= 0695161
| |
| }}
| |
| *{{Citation
| |
| | last=Rotman
| |
| | first=Joseph J.
| |
| | title=An introduction to the theory of groups
| |
| | publisher=Springer-Verlag
| |
| | year=1995
| |
| | edition=4
| |
| | isbn=978-0-387-94285-8
| |
| }}
| |
| * P.R. Girard (1984) "The quaternion group and modern physics", [[European Journal of Physics]] 5:25–32.
| |
| *{{Citation
| |
| | last=Hall
| |
| | first=Marshall
| |
| | title=The theory of groups
| |
| | publisher=AMS Bookstore
| |
| | year=1999
| |
| | edition=2
| |
| | isbn=0-8218-1967-4
| |
| }}
| |
| *{{Citation
| |
| | last=Kurosh
| |
| | first=Alexander G.
| |
| | title=Theory of Groups
| |
| | publisher=AMS Bookstore
| |
| | year=1979
| |
| | isbn=0-8284-0107-1
| |
| }}
| |
| | |
| ==External links==
| |
| * {{mathworld | urlname = QuaternionGroup | title = Quaternion group}}
| |
| | |
| [[Category:Group theory]]
| |
| [[Category:Finite groups]]
| |
| [[Category:Quaternions]]
| |
There are numerous proven methods to treat hemorrhoids which are powerful for lengthy expression hemorrhoid relief. Because, while hemorrhoids, like numerous other illnesses have a genetic component - when a mom or daddy had them you're more likely to receive them - they furthermore are influenced by lifestyle. Some of the factors which contribute to the occurrence of hemorrhoids are chronic constipation, sitting for extended periods of time, and a sofa potato lifestyle. So, hemorrhoids will be cured or at least place into significant remission by utilizing some good sense life-style techniques. I will focus on the easiest of these techniques to implement.
Another sort of stop bleeding hemorrhoids is the pill that is to be swallowed. It assists to regulate blood flows plus stress inside the body system. This helps to tighten vein tissues plus this delivers a relaxation for the hemorrhoid region. This nonetheless has certain side effects that may not augur well for some individuals. Pharmacists may actually not prescribe this option as a result of the negative effects.
Yes, hemorrhoids are painful, irritating and embarrassing. Folks wish a rapid fix for which. This is where the lotions come inside, plus they do function. Except they never stop the hemorrhoids from coming back. Would it not create more sense to find out what exactly is causing them and deal with which when and for all?
Lunch plus Dinner. Gradually add real fruits plus veggies to the meals, plus substitute entire grains for white flour and pasta for an additional fiber punch.
Well, he HAD heard about a hemorrhoids house remedy or 2, he mentioned, yet couldn't absolutely remember any details regarding them. I told him I needed time to consider the upcoming step and got from there plus into the bright sunlight because quick because I may. Surgery for hemorrhoids surely wasn't my first choice.
The operation procedures commonly involve scalpel, this is commonly done by cutting away the swelling and closing up the wounds. With this procedure you're required to stay inside the hospital following the procedure so which you will be provided some painkillers. Normally you'll feel the serious pain when the anesthesia wears off.
Piles may change each aspect of your lifetime, from how commonly we take a seat and curl about how much food you eat plus for many; it could feel like we can not do anything to create your hemorrhoids disappear. While many persons may have the periodic flare up, a lot of individuals have piles that have progressed into a long-term situation. As an example, many fresh mothers suffer from hemorrhoids from labour. The problem is the fact that based on the intensity, you can possibly end up suffering with this problem again, plus again plus again.