|
|
| Line 1: |
Line 1: |
| [[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.]]
| | This week, let's look at healthy weight ranges. Many dieters don't really learn what they must weigh and this usually creates a ideal deal of self-imposed strain plus pressure. It is important to learn what exactly is healthy for the height and to know the dangers plus advantages of each fat category in order to create educated decisions about a health. So let's make it a habit to refer to research whenever setting the goals. Bookmark this hub plus visit back any time to keep track of the fat category.<br><br>The primary sources of these "bad fats" are animal goods (meats, poultry, lard, butter), and certain oils, specifically the "tropical" oils, like palm plus coconut.<br><br>WHR is waist to height ratio assessed at home with a measuring tape. Divide waist circumference by the cool circumference. A ratio of .83 or less for guys and .71 or less for ladies is considered low risk (age 20-29).<br><br>Waking up this morning I was so hungry however I decided to stick with the diet. After my grapefruit breakfast, I am going on another bicycle ride. Well my belly hurt so much it was difficult to eat my lunch. I am thus hungry. I went over to my sisters home plus they you having pizza for dinner. I was tempted to eat it, yet I didnt. Now Im getting prepared to consume dinner. Im so happy for dinner because I feel which this might be the best dinner from the four days. I got a hamburger patty and stewed tomatoes. I then dipped the patty into the stewed tomatoes, it tasted like ketchup found on the hamburger. Tonights dinner had filled me up, which is good considering I dont need to go to bed with stomach cramps again. Im going to bed today plus it feels superior to be full.<br><br>Number 10. Wedding photographers, aside from being the creative monsters they are, should usually be approachable and easy to talk to. Why? Just to learn what the customer wants plus requires in the finish product. Remember, the images that you'll take are not just your precious works of art; it happens to be additionally the customers memoirs of 1 of the most unforgettable occasions of their lives. The photos should fit the clients taste. Communicate, communicate, communicate, be approachable with a handsome smile; besides, they wont pay we if they dont get what they desire.<br><br>BMI is usually expressed in metric values, yet [http://safedietplansforwomen.com/waist-to-height-ratio waist height ratio] this can be converted to standard units. Many resources, particularly web sites, have BMI calculators embedded in them yet they all use the same equation.<br><br>This puts me at the excellent end of regular. If I were to gain simply 10 pounds I would, according to BMI, be obese. Then most people tell me which I am skinny, which I think is form of silly. I am not skinny nevertheless I certainly do not believe which I would be obese at 185 pounds either. But, I do not like myself at which weight and I might not allow it to happen, however, which is my own individual problem. Of more value, both medically and to me personally, is that my percentage body fat runs about 14%, that is lower then many people of the same height and weight.<br><br>I was most satisfied plus impressed with the total results. The right thing for me today is the fact that I don't have to exercise all time to stay fit. I have absolutely invested enough time and cash following all those conventional techniques like going to gym and working out for hours. |
| | |
| 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]]
| |
| | |
| :<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 ==
| |
| | |
| 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> | |
| | |
| {|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
| |
| |-
| |
| ! −1
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| | −1 || 1 || −i || i || −j || j || −k || k
| |
| |-
| |
| ! i
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| | i || −i || −1 || 1 || k || −k || −j || j
| |
| |-
| |
| ! −i
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| | −i || i || 1 || −1 || −k || k || j || −j
| |
| |-
| |
| ! j
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| | j || −j || −k || k || −1 || 1 || i || −i
| |
| |-
| |
| ! −j
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| | −j || j || k || −k || 1 || −1 || −i || i
| |
| |-
| |
| ! k
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| | k || −k || j || −j || −i || i || −1 || 1
| |
| |-
| |
| ! −k
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| | −k || k || −j || j || i || −i || 1 || −1
| |
| |}
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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]].
| |
| | |
| :<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.
| |
| \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>
| |
| | |
| :<math>i \mapsto \begin{pmatrix}
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| i & 0 \\
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| 0 & -i
| |
| \end{pmatrix}</math>
| |
| | |
| :<math>j \mapsto \begin{pmatrix}
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| 0 & 1 \\
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| -1 & 0
| |
| \end{pmatrix}</math>
| |
| | |
| :<math>k \mapsto \begin{pmatrix}
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| 0 & i \\
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| i & 0
| |
| \end{pmatrix}</math>
| |
| | |
| 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
| |
| | |
| :<math>\mathrm{Q} = \{\pm 1, \pm i, \pm j, \pm k\} \to \mathrm{GL}(2,3)</math>
| |
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| is given by
| |
| | |
| :<math>1 \mapsto \begin{pmatrix}
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| 1 & 0 \\
| |
| 0 & 1
| |
| \end{pmatrix}</math>
| |
| | |
| :<math>i \mapsto \begin{pmatrix}
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| 1 & 1 \\
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| 1 & -1
| |
| \end{pmatrix}</math>
| |
| | |
| :<math>j \mapsto \begin{pmatrix}
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| -1 & 1 \\
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| 1 & 1
| |
| \end{pmatrix}</math>
| |
| | |
| :<math>k \mapsto \begin{pmatrix}
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| 0 & -1 \\
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| 1 & 0
| |
| \end{pmatrix}</math>
| |
| | |
| 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
| |
| :<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 | |
| | last = Dean | first = Richard
| |
| | authorlink = Richard Dean (mathematician)
| |
| | year = 1981
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| | title = A Rational Polynomial whose Group is the Quaternions
| |
| | journal = [[American Mathematical Monthly|The American Mathematical Monthly]]
| |
| | volume = 88 | issue = 1 | pages = 42–45
| |
| | jstor = 2320711
| |
| }}</ref>
| |
| | |
| ==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
| |
| | |
| :<math>\left(\begin{array}{cc}
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| \omega_n & 0 \\
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| 0 & \overline{\omega}_n
| |
| \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
| |
| \end{array}
| |
| \right)
| |
| </math>
| |
| | |
| 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}}.
| |
| | |
| 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'')}}.
| |
| | |
| The [[Brauer–Suzuki theorem]] shows that groups whose Sylow 2-subgroups are generalized quaternion cannot be simple. | |
| | |
| ==See also==
| |
| *[[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]]
| |
| *[[16-cell]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *{{citation
| |
| |author-link=Michael Artin
| |
| |last=Artin
| |
| |first=Michael
| |
| |title=Algebra
| |
| |publisher=Prentice Hall
| |
| |year=1991
| |
| |isbn=978-0-13-004763-2
| |
| }}
| |
| *{{citation
| |
| |last=Brown
| |
| |first=Kenneth S.|authorlink=Kenneth Brown (mathematician)
| |
| |title=Cohomology of groups
| |
| |publisher=Springer-Verlag
| |
| |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}}
| |
| * 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]]
| |
This week, let's look at healthy weight ranges. Many dieters don't really learn what they must weigh and this usually creates a ideal deal of self-imposed strain plus pressure. It is important to learn what exactly is healthy for the height and to know the dangers plus advantages of each fat category in order to create educated decisions about a health. So let's make it a habit to refer to research whenever setting the goals. Bookmark this hub plus visit back any time to keep track of the fat category.
The primary sources of these "bad fats" are animal goods (meats, poultry, lard, butter), and certain oils, specifically the "tropical" oils, like palm plus coconut.
WHR is waist to height ratio assessed at home with a measuring tape. Divide waist circumference by the cool circumference. A ratio of .83 or less for guys and .71 or less for ladies is considered low risk (age 20-29).
Waking up this morning I was so hungry however I decided to stick with the diet. After my grapefruit breakfast, I am going on another bicycle ride. Well my belly hurt so much it was difficult to eat my lunch. I am thus hungry. I went over to my sisters home plus they you having pizza for dinner. I was tempted to eat it, yet I didnt. Now Im getting prepared to consume dinner. Im so happy for dinner because I feel which this might be the best dinner from the four days. I got a hamburger patty and stewed tomatoes. I then dipped the patty into the stewed tomatoes, it tasted like ketchup found on the hamburger. Tonights dinner had filled me up, which is good considering I dont need to go to bed with stomach cramps again. Im going to bed today plus it feels superior to be full.
Number 10. Wedding photographers, aside from being the creative monsters they are, should usually be approachable and easy to talk to. Why? Just to learn what the customer wants plus requires in the finish product. Remember, the images that you'll take are not just your precious works of art; it happens to be additionally the customers memoirs of 1 of the most unforgettable occasions of their lives. The photos should fit the clients taste. Communicate, communicate, communicate, be approachable with a handsome smile; besides, they wont pay we if they dont get what they desire.
BMI is usually expressed in metric values, yet waist height ratio this can be converted to standard units. Many resources, particularly web sites, have BMI calculators embedded in them yet they all use the same equation.
This puts me at the excellent end of regular. If I were to gain simply 10 pounds I would, according to BMI, be obese. Then most people tell me which I am skinny, which I think is form of silly. I am not skinny nevertheless I certainly do not believe which I would be obese at 185 pounds either. But, I do not like myself at which weight and I might not allow it to happen, however, which is my own individual problem. Of more value, both medically and to me personally, is that my percentage body fat runs about 14%, that is lower then many people of the same height and weight.
I was most satisfied plus impressed with the total results. The right thing for me today is the fact that I don't have to exercise all time to stay fit. I have absolutely invested enough time and cash following all those conventional techniques like going to gym and working out for hours.