https://en.formulasearchengine.com/index.php?title=Transcranial_magnetic_stimulation&feed=atom&action=historyTranscranial magnetic stimulation - Revision history2024-03-29T06:40:48ZRevision history for this page on the wikiMediaWiki 1.42.0-wmf.5https://en.formulasearchengine.com/index.php?title=Transcranial_magnetic_stimulation&diff=285768&oldid=preven>2over0: /* top */ TMS does not redirect here - no need for hatnote2015-01-11T17:12:07Z<p><span dir="auto"><span class="autocomment">top: </span> <a href="/index.php?title=TMS&action=edit&redlink=1" class="new" title="TMS (page does not exist)">TMS</a> does not redirect here - no need for hatnote</span></p>
<a href="https://en.formulasearchengine.com/index.php?title=Transcranial_magnetic_stimulation&diff=285768&oldid=285767">Show changes</a>en>2over0https://en.formulasearchengine.com/index.php?title=Transcranial_magnetic_stimulation&diff=285767&oldid=preven>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9949)2014-02-25T21:19:19Z<p><a href="/index.php?title=WP:CHECKWIKI&action=edit&redlink=1" class="new" title="WP:CHECKWIKI (page does not exist)">WP:CHECKWIKI</a> error fix for #61. Punctuation goes before References. Do <a href="https://en.wikipedia.org/wiki/GENFIXES" class="extiw" title="wikipedia:GENFIXES">general fixes</a> if a problem exists. - using <a href="/index.php?title=Testwiki:AWB&action=edit&redlink=1" class="new" title="Testwiki:AWB (page does not exist)">AWB</a> (9949)</p>
<a href="https://en.formulasearchengine.com/index.php?title=Transcranial_magnetic_stimulation&diff=285767&oldid=2172">Show changes</a>en>BG19bothttps://en.formulasearchengine.com/index.php?title=Transcranial_magnetic_stimulation&diff=2172&oldid=preven>Talket: /* See also */ add electrical brain stimulation which is often compared with TMS in research, remove magnetic water treatment which has nothing to do with TMS2014-02-02T11:20:17Z<p><span dir="auto"><span class="autocomment">See also: </span> add electrical brain stimulation which is often compared with TMS in research, remove magnetic water treatment which has nothing to do with TMS</span></p>
<p><b>New page</b></p><div>[[File:Left cosets of Z 2 in Z 8.svg|thumb|G is the group <math>\mathbb{Z}/8\mathbb{Z}</math>, the [[Integers_modulo_n|integers mod 8]] under addition. The subgroup H contains only 0 and 4, and is isomorphic to <math>\mathbb{Z}/2\mathbb{Z}</math>. There are four left cosets of H: H itself, 1+H, 2+H, and 3+H (written using additive notation since this is the [[Ring (mathematics)#Definition and illustration|additive group]]). Together they partition the entire group G into equal-size, non-overlapping sets. The index [G : H] is 4.]]<br />
In [[mathematics]], if ''G'' is a [[group (mathematics)|group]], and ''H'' is a [[subgroup]] of ''G'', and ''g'' is an element of ''G'', then<br />
:''gH'' = {''gh'' : ''h'' an element of ''H''&thinsp;} is a '''left coset of ''H''''' in ''G'', and<br />
:''Hg'' = {''hg'' : ''h'' an element of ''H''&thinsp;} is a '''right coset of ''H''''' in ''G''.<br />
Only when ''H'' is [[normal subgroup|normal]] will the right and left cosets of ''H'' coincide, which is one definition of normality of a [[subgroup]]. Although derived from a subgroup, cosets are not usually themselves subgroups of ''G'', only subsets.<br />
<br />
A '''coset''' is a left or right coset of ''some'' [[subgroup]] in ''G''. Since ''Hg'' = ''g''&thinsp;(&thinsp;''g''<sup>−1</sup>''Hg''&thinsp;), the right coset ''Hg'' (of ''H'',&thinsp; with respect to ''g'') and the left coset ''g''&thinsp;(&thinsp;''g''<sup>−1</sup>''Hg''&thinsp;) (of the [[conjugacy class|conjugate]] subgroup ''g''<sup>−1</sup>''Hg''&thinsp;) are the same. Hence it is not meaningful to speak of a coset as being left or right unless one first specifies the underlying [[subgroup]]. In other words: a right coset of one subgroup equals a left coset of a different (conjugate) subgroup. If the left cosets and right cosets are the same then H is a [[normal subgroup]] and the cosets form a group called the [[quotient group|''quotient'' or ''factor'' group]].<br />
<br />
The map<br />
''gH''→(''gH'')<sup>−1</sup>=''Hg''<sup>−1</sup> defines a [[bijection]] between the left cosets and the right cosets of H, so the number of left cosets is equal to the number of right cosets. The common value is called the [[Index of a subgroup|index]] of ''H'' in ''G''.<br />
<br />
For [[abelian group]]s, left cosets and right cosets are always the same. If the group operation is written additively then the notation used changes to ''g''+''H'' or ''H''+''g''.<br />
<br />
Cosets are a basic tool in the study of groups; for example they play a central role in [[Lagrange's theorem (group theory)|Lagrange's theorem]].<br />
<br />
==Examples==<br />
==={-1,1}===<br />
Let ''G'' be the multiplicative group of {-1,1}, and ''H'' the trivial subgroup {1}. Then {-1} = (-1)''H'' = ''H''(-1) and {1} = 1''H'' = ''H''1 are the only cosets of ''H'' in ''G''. Because {1} is a normal subgroup of ''G'', the left and right cosets coincide and therefore do not need to be distinguished in this example.<br />
<br />
===Integers===<br />
Let ''G'' be the additive group of integers '''Z''' = {..., −2, −1, 0, 1, 2, ...} and ''H'' the subgroup ''m'''''Z''' = {..., −2''m'', −''m'', 0, ''m'', 2''m'', ...} where ''m'' is a positive integer. Then the cosets of ''H'' in ''G'' are the ''m'' sets ''m'''''Z''', ''m'''''Z'''+1, ... ''m'''''Z'''+(''m''−1), where ''m'''''Z'''+''a''={..., −2''m''+''a'', −''m''+''a'', ''a'', ''m''+''a'', 2''m''+''a'', ...}. There are no more than ''m'' cosets, because ''m'''''Z'''+''m''=''m''('''Z'''+1)=''m'''''Z'''. The coset ''m'''''Z'''+''a'' is the [[Modular arithmetic|congruence class]] of ''a'' modulo ''m''.<ref>Joshi p. 323</ref><br />
<br />
===Vectors===<br />
Another example of a coset comes from the theory of [[vector space]]s. The elements (vectors) of a vector space form an [[abelian group]] under [[vector addition]]. It is not hard to show that [[linear subspace|subspaces]] of a vector space are [[subgroups]] of this group. For a vector space ''V'', a subspace ''W'', and a fixed vector ''a'' in ''V'', the sets<br><br />
:<math>\{x \in V \colon x = a + n, n \in W\}</math><br />
are called [[affine subspace]]s, and are cosets (both left and right, since the group is abelian). In terms of [[Euclidean space|geometric]] vectors, these affine subspaces are all the "lines" or "planes" [[Parallel (geometry)|parallel]] to the subspace, which is a line or plane going through the origin.<br />
<br />
==Definition using equivalence classes==<br />
Some authors<ref>e.g. Zassenhaus</ref> define the left cosets of H in G to be the [[equivalence class]]es under the [[equivalence relation]] on ''G'' given by ''x''&nbsp;~&nbsp;''y'' if and only if ''x''<sup>−1</sup>''y'' ∈ ''H''. The relation can also be defined by ''x''&nbsp;~&nbsp;''y'' if and only if ''xh''=''y'' for some ''h'' in ''H''. It can be shown that the relation given is, in fact, an [[equivalence relation]] and that the two definitions are equivalent. It follows that any two left cosets of ''H'' in ''G'' are either identical or [[Disjoint sets|disjoint]]. In other words every element of ''G'' belongs to one and only one left coset and so the left cosets form a [[partition of a set|partition]] of ''G''.<ref>Joshi Corrolary 2.3</ref> Corresponding statements are true for right cosets.<br />
<br />
==Double cosets==<br />
{{main|Double coset}}<br />
Given two subgroups, ''H'' and ''K'' of a group ''G'', the '''double coset''' of ''H'' and ''K'' in ''G'' are sets of the form ''HgK'' = {''hgk'' : ''h'' an element of ''H '', ''k'' an element of ''K'' }. These are the left cosets of ''K'' and right cosets of ''H'' when ''H''=1 and ''K''=1 respectively.<ref>Scott p. 19</ref><br />
<br />
==Notation==<br />
<br />
Let ''G'' be a group with subgroups ''H'' and ''K.''<br />
#<math> G/H </math> denotes the set of left cosets <math>\{gH: g \in G\}</math> of ''H'' in ''G.''<br />
#<math> H\backslash G </math> denotes the set of right cosets <math>\{Hg: g \in G\}</math> of ''H'' in ''G.''<br />
#<math> K\backslash G/H </math> denotes the set of double cosets <math>\{KgH: g \in G\}</math> of ''H'' and ''K'' in ''G.''<br />
<br />
==General properties==<br />
The identity is in precisely one left or right coset, namely ''H'' itself. Thus ''H'' is both a left and right coset of itself.<br />
<br />
A '''coset representative''' is a representative in the equivalence class sense. A set of representatives of all the cosets is called a [[Transversal (combinatorics)|transversal]]. There are other types of equivalence relations in a group, such as conjugacy, that form different classes which do not have the properties discussed here.<br />
<br />
===Index of a subgroup===<br />
{{Main|Index of a subgroup}}<br />
All left cosets and all right cosets have the same [[order (group theory)|order]] (number of elements, or [[cardinality]] in the case of an [[Infinity|infinite]] ''H''), equal to the order of ''H'' (because ''H'' is itself a coset). Furthermore, the number of left cosets is equal to the number of right cosets and is known as the '''index''' of ''H'' in ''G'', written as [''G'' : ''H''&thinsp;]. [[Lagrange's theorem (group theory)|Lagrange's theorem]] allows us to compute the index in the case where ''G'' and ''H'' are finite, as per the formula:<br />
:|''G''&thinsp;| = [''G'' : ''H''&thinsp;] · |''H''&thinsp;|.<br />
This equation also holds in the case where the groups are infinite, although the meaning may be less clear.<br />
<br />
===Cosets and normality===<br />
If ''H'' is not [[normal subgroup|normal]] in ''G'', then its left cosets are different from its right cosets. That is, there is an ''a'' in ''G'' such that no element ''b'' satisfies ''aH'' = ''Hb''. This means that the partition of ''G'' into the left cosets of ''H'' is a different partition than the partition of ''G'' into right cosets of ''H''. (It is important to note that ''some'' cosets may coincide. For example, if ''a'' is in the [[center (group theory)|center]] of ''G'', then ''aH'' = ''Ha''.)<br />
<br />
On the other hand, the subgroup ''N'' is normal if and only if ''gN'' = ''Ng'' for all ''g'' in ''G''. In this case, the set of all cosets form a group called the [[quotient group]] ''G''&nbsp;/&nbsp;''N'' with the operation ∗ defined by (''aN''&thinsp;)∗(''bN''&thinsp;) = ''abN''. Since every right coset is a left coset, there is no need to differentiate "left cosets" from "right cosets".<br />
<br />
==Applications==<br />
*Cosets of '''Q''' in '''R''' are used in the construction of [[Vitali set]]s, a type of [[non-measurable set]].<br />
*Cosets are central in the definition of the [[Transfer (group theory)|transfer]].<br />
*Cosets are important in computational group theory. For example [[Optimal solutions for Rubik's Cube#Thistlethwaite's algorithm|Thistlethwaite's algorithm]] for solving [[Rubik's Cube]] relies heavily on cosets.<br />
*[[Coset leader]]s are used in decoding received data in [[Linear code|linear error-correcting codes]].<br />
<br />
==See also==<br />
*[[Double coset]]<br />
*[[Heap (mathematics)|Heap]]<br />
*[[Lagrange's theorem (group theory)|Lagrange's theorem]]<br />
*[[Quotient group]]<br />
*[[Coset enumeration]]<br />
<br />
==References==<br />
{{reflist}}<br />
<br />
*{{cite book|title=Group Theory|first=W.R.|last=Scott|publisher=Courier Dover Publications<br />
|year=1987|isbn=0-486-65377-3|chapter=§1.7 Cosets and index|pages=19 ff.}}<br />
*{{cite book|title=Foundations of Discrete Mathematics<br />
|first=K. D.|last=Joshi|publisher=New Age International|year=1989|isbn=81-224-0120-1<br />
|chapter=§5.2 Cosets of Subgroups|pages=322 ff.}}<br />
*{{cite book |title=The Theory of Groups<br />
|first=Hans J.|last=Zassenhaus|publisher=Courier Dover Publications|year=1999|isbn=0-486-40922-8<br />
|chapter=§1.4 Subgroups|pages=10 ff.|authorlink=Hans Zassenhaus}}<br />
<br />
==External links==<br />
*{{MathWorld|title=Coset|urlname=Coset|author=Nicolas Bray}}<br />
*{{MathWorld|title=Left Coset|urlname=LeftCoset}}<br />
*{{MathWorld|title=Right Coset|urlname=RightCoset}}<br />
*{{springer|title=Coset in a group|id=C/c026620|last=Ivanova|first=O.A.}}<br />
*{{PlanetMath|urlname=Coset|title=Coset}}<br />
*[http://www.millersville.edu/~bikenaga/abstract-algebra-1/coset/coset.html Illustrated examples]<br />
*{{cite web| publisher=The Group Properties Wiki| work=groupprops|url=http://groupprops.subwiki.org/wiki/Coset| title=Coset}}<br />
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[[Category:Group theory]]<br />
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[[de:Gruppentheorie#Nebenklassen]]<br />
[[ru:Глоссарий теории групп#К]]</div>en>Talket