|
|
| Line 1: |
Line 1: |
| {{For|the number of labeled trees in graph theory|Cayley's formula}}
| | BMI Basics: The Body Mass Index is worthwhile because a ballpark measurement. Health experts plus organizations like the WHO, employ it to gain a simple idea of whether a individual is obese, underweight, or is at a healthy fat, given their height. If you're my age (that I'm not about to disclose,) you were possibly introduced to the BMI in Middle School during PE. Back then I was taught to measure my BMI with a simple formula. A few punches found on the calculator and voila, my body fat was quantified.<br><br>The target fat could nonetheless become the weight based on the standard BMI height and weight charts. The range presented in the [http://safedietplans.com/bmi-chart bmi chart] is reasonable, and even with muscle along with a big body frame, women must be capable to reach the healthy range. Moreover, women are allowed to be a little heavier because they grow older.<br><br>37. Making time for yourself: Running = sanity. Alone or with neighbors it has great therapeutic results that last all day bmi chart men. I find doing it early each morning is best as I understand I'll receive my run inside and "life stuff" during the day will not receive inside the method.<br><br>Rest: This has become the most significant parts of my training. If I don't get enough rest, my body begins to break down. Listen (fairly closely) to the body.<br><br>Having sex using the missionary position assist to get pregnant- the time-proven 'man-on-top' position works with gravity to motivate semen flow toward the uterus to maximize the opportunity of a sperm uniting with the egg plus causing conception. One of the connected best tricks on getting expecting is -- after ejaculation, the girl could stay on her back with her legs bent a few minutes to further maximize semen flow toward the uterus.<br><br>Dont do anything extreme whenever you are trying bmi chart women to take off fat. Eat sensibly, omitting 500 calories a day for a weight reduction of one pound per week. Exercise, slowly adding more repetitions for much quicker weight reduction. You can be doing a heart a favor.<br><br>That sounds actually complicated, yet when we have the numbers and are making the actual calculations, it's not nearly as hard as it sounds, although you certainly need a calculator of some sort. This really is how men should measure their body fat. For women, the measurements are different.<br><br>If you think you are general weight overweight, the number one thing you can do is consult your doctor, and begin to make healthier options inside your existence. For more information feel free to go and visit a few of the links I've provided. |
| | |
| In [[group theory]], '''Cayley's theorem''', named in honor of [[Arthur Cayley]], states that every [[group (mathematics)|group]] ''G'' is [[group isomorphism|isomorphic]] to a [[subgroup]] of the [[symmetric group]] acting on ''G''.<ref>Jacobson (2009), p. 38.</ref> This can be understood as an example of the [[group action]] of ''G'' on the elements of ''G''.<ref>Jacobson (2009), p. 72, ex. 1.</ref>
| |
| | |
| A [[permutation]] of a set ''G'' is any [[bijective]] [[function (mathematics)|function]] taking ''G'' onto ''G''; and the set of all such functions forms a group under [[function composition]], called ''the symmetric group on'' ''G'', and written as Sym(''G'').<ref>Jacobson (2009), p. 31.</ref>
| |
| | |
| Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as ('''''R''''',+)) as a [[permutation group]] of some underlying set. Thus, theorems which are true for permutation groups are true for groups in general.
| |
| | |
| == History ==
| |
| Although Burnside<ref>{{Citation | last = Burnside | first = William | author-link = William Burnside | title = Theory of Groups of Finite Order | location = Cambridge | year = 1911 | edition = 2 | isbn = 0-486-49575-2}}</ref>
| |
| attributes the theorem
| |
| to Jordan,<ref>{{Citation | last = Jordan | first = Camille | author-link = Camille Jordan | title = Traite des substitutions et des equations algebriques | publisher = Gauther-Villars | location = Paris | year = 1870}}</ref>
| |
| Eric Nummela<ref>{{Citation | last = Nummela | first = Eric | title = Cayley's Theorem for Topological Groups | journal = American Mathematical Monthly | volume = 87 | issue = 3 | year = 1980 | pages = 202–203 | doi = 10.2307/2321608 | jstor = 2321608 | publisher = Mathematical Association of America}}</ref>
| |
| nonetheless argues that the standard name—"Cayley's Theorem"—is in fact appropriate. Cayley, in his original 1854 paper,<ref>{{Citation | last = Cayley | first = Arthur | author-link = Arthur Cayley | title = On the theory of groups as depending on the symbolic equation θ<sup>n</sup>=1 | journal = Philosophical Magazine | volume = 7 | issue = 42 | pages = 40–47 | year = 1854 | url = http://books.google.com/books?id=_LYConosISUC&pg=PA40#v=onepage&q&f=false }}</ref>
| |
| showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an isomorphism). However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so.
| |
| | |
| == Proof of the theorem ==
| |
| Where ''g'' is any element of a group ''G'' with operation ∗, consider the function {{nowrap|''f''<sub>''g''</sub> : ''G'' → ''G''}}, defined by {{nowrap|1=''f''<sub>''g''</sub>(''x'') = ''g'' ∗ ''x''}}. By the existence of inverses, this function has a two-sided inverse, <math>f_{g^{-1}}</math>. So multiplication by ''g'' acts as a [[bijective]] function. Thus, ''f''<sub>''g''</sub> is a permutation of ''G'', and so is a member of Sym(''G'').
| |
| | |
| The set {{nowrap|1=''K'' = {''f''<sub>''g''</sub> : ''g'' ∈ ''G''} }} is a subgroup of Sym(''G'') that is isomorphic to ''G''. The fastest way to establish this is to consider the function {{nowrap|''T'' : ''G'' → Sym(''G'')}} with {{nowrap|1=''T''(''g'') = ''f''<sub>''g''</sub>}} for every ''g'' in ''G''. ''T'' is a [[group homomorphism]] because (using · to denote composition in Sym(''G'')):
| |
| | |
| :<math> (f_g \cdot f_h)(x) = f_g(f_h(x)) = f_g(h*x) = g*(h*x) = (g*h)*x = f_{g*h}(x) ,</math>
| |
| for all ''x'' in ''G'', and hence:
| |
| :<math> T(g) \cdot T(h) = f_g \cdot f_h = f_{g*h} = T(g*h) .</math>
| |
| The homomorphism ''T'' is also [[injective]] since {{nowrap|1=''T''(''g'') = id<sub>''G''</sub>}} (the identity element of Sym(''G'')) implies that {{nowrap|1=''g'' ∗ ''x'' = ''x''}} for all ''x'' in ''G'', and taking ''x'' to be the identity element ''e'' of ''G'' yields {{nowrap|1=''g'' = ''g'' ∗ ''e'' = ''e''}}. Alternatively, ''T'' is also [[injective]] since, if {{nowrap|1=''g'' ∗ ''x'' = ''g''′ ∗ ''x''}} implies that {{nowrap|1=''g'' = ''g''′}} (because every group is [[cancellative]]).
| |
| | |
| Thus ''G'' is isomorphic to the image of ''T'', which is the subgroup ''K''.
| |
| | |
| ''T'' is sometimes called the ''regular representation of'' ''G''.
| |
| | |
| === Alternative setting of proof ===
| |
| An alternative setting uses the language of [[group action]]s. We consider the group <math>G</math> as a G-set, which can be shown to have permutation representation, say <math>\phi</math>.
| |
| | |
| Firstly, suppose <math>G=G/H</math> with <math>H=\{e\}</math>. Then the group action is <math>g.e</math> by [[group action|classification of G-orbits]] (also known as the orbit-stabilizer theorem).
| |
| | |
| Now, the representation is faithful if <math>\phi</math> is injective, that is, if the kernel of <math>\phi</math> is trivial. Suppose <math>g\in\ker\phi</math> Then, <math>g=g.e=\phi(g).e</math> by the equivalence of the permutation representation and the group action. But since <math>g \in \ker\phi</math>, <math>\phi(g)=e</math> and thus <math>\ker\phi</math> is trivial. Then <math>\mathrm{Im} \phi < G</math> and thus the result follows by use of the [[first isomorphism theorem]].
| |
| | |
| ==Remarks on the regular group representation==
| |
| The identity group element corresponds to the identity permutation. All other group elements correspond to a permutation that does not leave any element unchanged. Since this also applies for powers of a group element, lower than the order of that element, each element corresponds to a permutation which consists of cycles which are of the same length: this length is the order of that element. The elements in each cycle form a left [[coset]] of the subgroup generated by the element.
| |
| | |
| ==Examples of the regular group representation==
| |
| Z<sub>2</sub> = {0,1} with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12). E.g. 0 +1 = 1 and 1+1 = 0 , so 1 -> 0 and 0 -> 1, as they would under a permutation.
| |
| | |
| Z<sub>3</sub> = {0,1,2} with addition modulo 3; group element 0 corresponds to the identity permutation e, group element 1 to permutation (123), and group element 2 to permutation (132). E.g. 1 + 1 = 2 corresponds to (123)(123)=(132).
| |
| | |
| Z<sub>4</sub> = {0,1,2,3} with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432).
| |
| | |
| The elements of [[Klein four-group]] {e, a, b, c} correspond to e, (12)(34), (13)(24), and (14)(23).
| |
| | |
| S<sub>3</sub> ([[dihedral group of order 6]]) is the group of all permutations of 3 objects, but also a permutation group of the 6 group elements:
| |
| | |
| <!-- Looks ugly if it's left-aligned/non-square cells, so a bit of customization is good here -->
| |
| {| class="wikitable" style="text-align: center;"
| |
| ! style="width: 1.5em; height: 1.5em;" | *
| |
| ! style="width: 1.5em;" | ''e''
| |
| ! style="width: 1.5em;" | ''a''
| |
| ! style="width: 1.5em;" | ''b''
| |
| ! style="width: 1.5em;" | ''c''
| |
| ! style="width: 1.5em;" | ''d''
| |
| ! style="width: 1.5em;" | ''f''
| |
| ! permutation
| |
| |-
| |
| ! style="height: 1.5em;" | ''e''
| |
| | ''e'' || ''a'' || ''b'' || ''c'' || ''d'' || ''f'' || ''e''
| |
| |-
| |
| ! style="height: 1.5em;" | ''a''
| |
| | ''a'' || ''e'' || ''d'' || ''f'' || ''b'' || ''c'' || (12)(35)(46)
| |
| |-
| |
| ! style="height: 1.5em;" | ''b''
| |
| | ''b'' || ''f'' || ''e'' || ''d'' || ''c'' || ''a'' || (13)(26)(45)
| |
| |-
| |
| ! style="height: 1.5em;" | ''c''
| |
| | ''c'' || ''d'' || ''f'' || ''e'' || ''a'' || ''b'' || (14)(25)(36)
| |
| |-
| |
| ! style="height: 1.5em;" | ''d''
| |
| | ''d'' || ''c'' || ''a'' || ''b'' || ''f'' || ''e'' || (156)(243)
| |
| |-
| |
| ! style="height: 1.5em;" | ''f''
| |
| | ''f'' || ''b'' || ''c'' || ''a'' || ''e'' || ''d'' || (165)(234)
| |
| |}
| |
| | |
| == See also ==
| |
| * [[Containment order]], a similar result in order theory
| |
| * [[Frucht's theorem]], every group is the automorphism group of a graph
| |
| * [[Yoneda lemma]], an analogue of Cayley's theorem in category theory
| |
| * [[representation theorem]]
| |
| | |
| == Notes ==
| |
| {{Reflist}}
| |
| | |
| == References ==
| |
| * {{Citation| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| year=2009| title=Basic algebra| edition=2nd| series= | publisher=Dover| isbn = 978-0-486-47189-1}}.
| |
| | |
| [[Category:Permutations]]
| |
| [[Category:Theorems in group theory]]
| |
| [[Category:Articles containing proofs]]
| |
BMI Basics: The Body Mass Index is worthwhile because a ballpark measurement. Health experts plus organizations like the WHO, employ it to gain a simple idea of whether a individual is obese, underweight, or is at a healthy fat, given their height. If you're my age (that I'm not about to disclose,) you were possibly introduced to the BMI in Middle School during PE. Back then I was taught to measure my BMI with a simple formula. A few punches found on the calculator and voila, my body fat was quantified.
The target fat could nonetheless become the weight based on the standard BMI height and weight charts. The range presented in the bmi chart is reasonable, and even with muscle along with a big body frame, women must be capable to reach the healthy range. Moreover, women are allowed to be a little heavier because they grow older.
37. Making time for yourself: Running = sanity. Alone or with neighbors it has great therapeutic results that last all day bmi chart men. I find doing it early each morning is best as I understand I'll receive my run inside and "life stuff" during the day will not receive inside the method.
Rest: This has become the most significant parts of my training. If I don't get enough rest, my body begins to break down. Listen (fairly closely) to the body.
Having sex using the missionary position assist to get pregnant- the time-proven 'man-on-top' position works with gravity to motivate semen flow toward the uterus to maximize the opportunity of a sperm uniting with the egg plus causing conception. One of the connected best tricks on getting expecting is -- after ejaculation, the girl could stay on her back with her legs bent a few minutes to further maximize semen flow toward the uterus.
Dont do anything extreme whenever you are trying bmi chart women to take off fat. Eat sensibly, omitting 500 calories a day for a weight reduction of one pound per week. Exercise, slowly adding more repetitions for much quicker weight reduction. You can be doing a heart a favor.
That sounds actually complicated, yet when we have the numbers and are making the actual calculations, it's not nearly as hard as it sounds, although you certainly need a calculator of some sort. This really is how men should measure their body fat. For women, the measurements are different.
If you think you are general weight overweight, the number one thing you can do is consult your doctor, and begin to make healthier options inside your existence. For more information feel free to go and visit a few of the links I've provided.