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In [[number theory]] '''Euler's criterion''' is a formula for determining whether an [[integer]] is a [[quadratic residue]] [[modular arithmetic|modulo]] a [[prime number|prime]]. Precisely,
{{For|the number of labeled trees in graph theory|Cayley's formula}}


Let ''p'' be an [[odd number|odd]] prime and ''a'' an integer [[coprime]] to ''p''. Then<ref>Gauss, DA, Art. 106</ref>
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>


:<math>
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>
a^{\tfrac{p-1}{2}} \equiv
\begin{cases}
\;\;\,1\pmod{p}& \text{ if there is an integer }x \text{ such that }a\equiv x^2 \pmod{p}\\
    -1\pmod{p}& \text{ if there is no such integer.}
\end{cases}
</math>


Euler's criterion can be concisely reformulated using the [[Legendre symbol]]:<ref>Hardy & Wright, thm. 83</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.
:<math>
\left(\frac{a}{p}\right) \equiv a^{(p-1)/2} \pmod p.
</math>


The criterion first appeared in a 1748 paper by [[Leonhard Euler|Euler]].<ref>Lemmermeyer, p. 4 cites two papers, E134 and E262 in the Euler Archive</ref>
== 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&mdash;"Cayley's Theorem"&mdash;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 = Phil. Mag. | volume = 7 | issue = 4 | pages = 40–47 | year = 1854}}</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==
== Proof of the theorem ==
Where ''g'' is any element of ''G'', consider the function ''f''<sub>''g''</sub> : ''G'' → ''G'', defined by ''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 proof uses fact that the residue classes modulo a prime number are a [[finite field|field]]. See the article [[Characteristic_(algebra)#Case_of_fields|prime field]] for more details. The fact that there are (''p'' − 1)/2 quadratic residues and the same number of nonresidues (mod ''p'') is proved in the article [[quadratic residue]].
The set <math>K = \{f_g: g \in G\}</math> is a subgroup of Sym(''G'') which is isomorphic to ''G''. The fastest way to establish this is to consider the function ''T'' : ''G'' → Sym(''G'') with ''T''(''g'') = ''f''<sub>''g''</sub> for every ''g'' in ''G''. ''T'' is a [[group homomorphism]] because (using "•" for composition in Sym(''G'')):


[[Fermat's little theorem]] says that
:<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)\mbox{,}</math>
:<math>
for all ''x'' in ''G'', and hence:
a^{p-1}\equiv 1 \pmod p.
:<math> T(g) \cdot T(h) = f_g \cdot f_h = f_{(g*h)} = T(g*h).</math>
</math>
The homomorphism ''T'' is also [[injective]] since ''T''(''g'') = id<sub>''G''</sub> (the identity element of Sym(''G'')) implies that ''g*x'' = ''x'' for all ''x'' in ''G'', and taking ''x'' to be the identity element ''e'' of ''G'' yields ''g'' = ''g''*''e'' = ''e''. Alternatively, ''T'' is also [[injective]] since, if ''g''*''x''=''g'''*''x'' implies ''g''=''g' '' (by post-multiplying with the inverse of ''x'', which exists because ''G'' is a group).
This can be written as
:<math>
(a^{\tfrac{p-1}{2}}-1)(a^{\tfrac{p-1}{2}}+1)\equiv 0 \pmod p.
</math>  
Since the integers mod ''p'' form
a field, one or the other of these factors must be congruent to zero.


Now if ''a'' is a quadratic residue, ''a'' &equiv; ''x''<sup>2</sup>,
Thus ''G'' is isomorphic to the image of ''T'', which is the subgroup ''K''.
:<math>
a^{\tfrac{p-1}{2}}\equiv{x^2}^{\tfrac{p-1}{2}}\equiv x^{p-1}\equiv1\pmod p.
</math>
So every quadratic residue (mod ''p'') makes the first factor zero.


[[Lagrange's theorem (number theory)|Lagrange's theorem]] says that there can be no more than (''p''&nbsp;−&nbsp;1)/2 values of ''a'' that make the first factor zero. But it is known that there are (''p''&nbsp;−&nbsp;1)/2 distinct quadratic residues (mod ''p''). Therefore they are precisely the residue classes that make the first factor zero. The other (''p''&nbsp;−&nbsp;1)/2 residue classes, the nonresidues, must be the ones making the second factor zero. This is Euler's criterion.
''T'' is sometimes called the ''regular representation of'' ''G''.


==Examples==
=== Alternative setting of proof ===
'''Example 1: Finding primes for which ''a'' is a residue'''
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>.


Let ''a'' = 17. For which primes ''p'' is 17 a quadratic residue?
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).


We can test prime ''p'''s manually given the formula above.
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]].


In one case, testing ''p'' = 3, we have 17<sup>(3 − 1)/2</sup> = 17<sup>1</sup> ≡ 2 ≡ −1 (mod 3), therefore 17 is not a quadratic residue modulo 3.
==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.


In another case, testing ''p'' = 13, we have 17<sup>(13 − 1)/2</sup> = 17<sup>6</sup> ≡ 1 (mod 13), therefore 17 is a quadratic residue modulo 13.  As confirmation, note that 17 ≡ 4 (mod 13), and 2<sup>2</sup> = 4.
==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).


We can do these calculations faster by using various modular arithmetic and Legendre symbol properties.
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).


If we keep calculating the values, we find:
Z<sub>4</sub> = {0,1,2,3} with addition modulo 4; the elements correspond to e, (1234), (13)(24), (1432).
:(17/''p'') = +1 for ''p'' = {13, 19, ...} (17 is a quadratic residue modulo these values)


:(17/''p'') = −1 for ''p'' = {3, 5, 7, 11, 23, ...} (17 is not a quadratic residue modulo these values).
The elements of [[Klein four-group]] {e, a, b, c} correspond to e, (12)(34), (13)(24), and (14)(23).


'''Example 2: Finding residues given a prime modulus ''p'' '''
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:


Which numbers are squares modulo 17 (quadratic residues modulo 17)?
<!-- 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)
|}


We can manually calculate:
== See also ==
: 1<sup>2</sup> = 1
* [[Containment order]], a similar result in order theory
: 2<sup>2</sup> = 4
* [[Frucht's theorem]], every group is the automorphism group of a graph
: 3<sup>2</sup> = 9
* [[Yoneda lemma]], an analogue of Cayley's theorem in category theory
: 4<sup>2</sup> = 16
* [[representation theorem]]
: 5<sup>2</sup> = 25 ≡ 8 (mod 17)
: 6<sup>2</sup> = 36 ≡ 2 (mod 17)
: 7<sup>2</sup> = 49 ≡ 15 (mod 17)
: 8<sup>2</sup> = 64 ≡ 13 (mod 17).


So the set of the quadratic residues modulo 17 is {1,2,4,8,9,13,15,16}.  Note that we did not need to calculate squares for the values 9 through 16, as they are all negatives of the previously squared values (e.g. 9 ≡ −8 (mod 17), so 9<sup>2</sup> ≡ (−8)<sup>2</sup> = 64 ≡ 13 (mod 17)).
== Notes ==
{{Reflist}}


We can find quadratic residues or verify them using the above formula.  To test if 2 is a quadratic residue modulo 17, we calculate 2<sup>(17 − 1)/2</sup> = 2<sup>8</sup> ≡ 1 (mod 17), so it is a quadratic residue.  To test if 3 is a quadratic residue modulo 17, we calculate 3<sup>(17 − 1)/2</sup> = 3<sup>8</sup> ≡ 16 ≡ −1 (mod 17), so it is not a quadratic residue.
== 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}}.


Euler's criterion is related to the [[Quadratic reciprocity|Law of quadratic reciprocity]] and is used in a definition of [[Euler–Jacobi pseudoprime]]s.
[[Category:Permutations]]
 
[[Category:Theorems in group theory]]
==See also==
 
==Notes==
 
{{reflist}}
 
==References==
 
The ''[[Disquisitiones Arithmeticae]]'' has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
 
 
*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | last2 = Clarke | first2 = Arthur A. (translator into English) 
  | title = Disquisitiones Arithemeticae (Second, corrected edition)
  | publisher = [[Springer Science+Business Media|Springer]]
  | location = New York
  | date = 1986
  | isbn = 0-387-96254-9}}
 
*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | last2 = Maser | first2 = H. (translator into German) 
  | title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
  | publisher = Chelsea
  | location = New York
  | date = 1965
  | isbn = 0-8284-0191-8}}
 
 
*{{citation
  | last1 = Hardy  | first1 = G. H.
  | last2 = Wright | first2 = E. M.
  | title = An Introduction to the Theory of Numbers (Fifth edition)
  | publisher = [[Oxford University Press]]
  | location = Oxford
  | date = 1980
  | isbn = 978-0-19-853171-5}}
 
*{{citation
  | last1 = Lemmermeyer  | first1 = Franz
  | title = Reciprocity Laws: from Euler to Eisenstein
  | publisher = [[Springer Science+Business Media|Springer]]
  | location = Berlin
  | date = 2000
  | isbn = 3-540-66957-4}}
 
==External links==
 
*[http://www.math.dartmouth.edu/~euler/index.html The Euler Archive]
 
{{DEFAULTSORT:Euler's Criterion}}
[[Category:Modular arithmetic]]
[[Category:Articles containing proofs]]
[[Category:Articles containing proofs]]
[[Category:Quadratic residue]]
[[Category:Theorems about prime numbers]]


[[ca:Criteri d'Euler]]
[[de:Satz von Cayley]]
[[es:Criterio de Euler]]
[[es:Teorema de Cayley]]
[[fr:Critère d'Euler]]
[[fr:Théorème de Cayley]]
[[it:Criterio di Eulero]]
[[it:Teorema di Cayley]]
[[he:מבחן אוילר]]
[[he:משפט קיילי]]
[[pl:Kryterium Eulera]]
[[hu:Cayley-tétel]]
[[sv:Eulers kriterium]]
[[nl:Stelling van Cayley]]
[[zh:欧拉准则]]
[[pl:Twierdzenie Cayleya]]
[[ru:Теорема Кэли (теория групп)]]
[[fi:Cayleyn lause]]
[[sv:Cayleys sats]]
[[uk:Теорема Келі (теорія груп)]]
[[zh:凱萊定理]]

Revision as of 02:42, 10 August 2014

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.

In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G.[1] This can be understood as an example of the group action of G on the elements of G.[2]

A permutation of a set G is any bijective 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).[3]

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[4] attributes the theorem to Jordan,[5] Eric Nummela[6] nonetheless argues that the standard name—"Cayley's Theorem"—is in fact appropriate. Cayley, in his original 1854 paper,[7] 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 G, consider the function fg : G → G, defined by fg(x) = g*x. By the existence of inverses, this function has a two-sided inverse, fg−1. So multiplication by g acts as a bijective function. Thus, fg is a permutation of G, and so is a member of Sym(G).

The set K={fg:g∈G} is a subgroup of Sym(G) which is isomorphic to G. The fastest way to establish this is to consider the function T : G → Sym(G) with T(g) = fg for every g in G. T is a group homomorphism because (using "•" for composition in Sym(G)):

(fg⋅fh)(x)=fg(fh(x))=fg(h*x)=g*(h*x)=(g*h)*x=f(g*h)(x),

for all x in G, and hence:

T(g)⋅T(h)=fg⋅fh=f(g*h)=T(g*h).

The homomorphism T is also injective since T(g) = idG (the identity element of Sym(G)) implies that g*x = x for all x in G, and taking x to be the identity element e of G yields g = g*e = e. Alternatively, T is also injective since, if g*x=g'*x implies g=g' (by post-multiplying with the inverse of x, which exists because G is a group).

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 actions. We consider the group G as a G-set, which can be shown to have permutation representation, say ϕ.

Firstly, suppose G=G/H with H={e}. Then the group action is g.e by classification of G-orbits (also known as the orbit-stabilizer theorem).

Now, the representation is faithful if ϕ is injective, that is, if the kernel of ϕ is trivial. Suppose g∈ker⁡ϕ Then, g=g.e=ϕ(g).e by the equivalence of the permutation representation and the group action. But since g∈ker⁡ϕ, ϕ(g)=e and thus ker⁡ϕ is trivial. Then Imϕ<G 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

Z2 = {0,1} with addition modulo 2; group element 0 corresponds to the identity permutation e, group element 1 to permutation (12).

Z3 = {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).

Z4 = {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).

S3 (dihedral group of order 6) is the group of all permutations of 3 objects, but also a permutation group of the 6 group elements:

* e a b c d f permutation
e e a b c d f e
a a e d f b c (12)(35)(46)
b b f e d c a (13)(26)(45)
c c d f e a b (14)(25)(36)
d d c a b f e (156)(243)
f f b c a e d (165)(234)

See also

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

  • 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.

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    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.

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de:Satz von Cayley es:Teorema de Cayley fr:Théorème de Cayley it:Teorema di Cayley he:משפט קיילי hu:Cayley-tétel nl:Stelling van Cayley pl:Twierdzenie Cayleya ru:Теорема Кэли (теория групп) fi:Cayleyn lause sv:Cayleys sats uk:Теорема Келі (теорія груп) zh:凱萊定理

  1. ↑ Jacobson (2009), p. 38.
  2. ↑ Jacobson (2009), p. 72, ex. 1.
  3. ↑ Jacobson (2009), p. 31.
  4. ↑ 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
  5. ↑ 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
  6. ↑ 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
  7. ↑ 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
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