Groupprops:Definition - Revision history

Vipul: /* Equivalent definitions as a list */

2008-06-20T11:15:20Z

Equivalent definitions as a list

← Older revision		Revision as of 11:15, 20 June 2008
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	We try to follow these conventions:		We try to follow these conventions:

	* When multiple definitions are given, the same order is maintained for the definitions in the symbol-free ~~frmat~~ and the definition with symbols format.		* When multiple definitions are given, the same order is maintained for the definitions in the symbol-free format and the definition with symbols format.
	* The multiple definitions should preferably be numbered, making it easier to refer to the numbering		* The multiple definitions should preferably be numbered, making it easier to refer to the numbering
	* The definitions are ordered in a way that makes the equivalence between them as self-evident as possible.		* The definitions are ordered in a way that makes the equivalence between them as self-evident as possible.

Vipul: /* Defining ingredients */

2008-05-29T00:31:10Z

Defining ingredients

← Older revision		Revision as of 00:31, 29 May 2008
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	The wiki markup contains information about what terms are used as ingredients in defining a particular term (using semantic MediaWiki). At the bottom of the article, a box called ''Facts'' gives a list of defining ingredients. Clicking on the magnifying glass icon next to any of these ingredients gives a list of all the other terms that use the same defining ingredient.		The wiki markup contains information about what terms are used as ingredients in defining a particular term (using semantic MediaWiki). At the bottom of the article, a box called ''Facts'' gives a list of defining ingredients. Clicking on the magnifying glass icon next to any of these ingredients gives a list of all the other terms that use the same defining ingredient.

			===Links to survey articles clarifying the definition===

			For definitions of important terms, the definition section may begin with a box containing links to survey articles clarifying the definition.

	==Other comments==		==Other comments==

	The definition section may also contain a few other comments, in so far as these comments play a direct role in clarifying the definition. Examples are typically reserved for another section, and under special circumstances, there is a separate section titled ''Importance''.		The definition section may also contain a few other comments, in so far as these comments play a direct role in clarifying the definition. Examples are typically reserved for another section, and under special circumstances, there is a separate section titled ''Importance''.

Vipul: /* Locating defining ingredients */

2008-05-11T23:18:14Z

Locating defining ingredients

← Older revision		Revision as of 23:18, 11 May 2008
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	===Locating defining ingredients===		===Locating defining ingredients===

	The wiki markup contains information about what terms are used as ingredients in defining a particular term (using semantic MediaWiki). At the bottom of the article, a box called ''Facts'' gives a list of defining ingredients. ~~Clocking~~ on the magnifying glass icon next to any of these ingredients gives a list of all the other terms that use the same defining ingredient.		The wiki markup contains information about what terms are used as ingredients in defining a particular term (using semantic MediaWiki). At the bottom of the article, a box called ''Facts'' gives a list of defining ingredients. Clicking on the magnifying glass icon next to any of these ingredients gives a list of all the other terms that use the same defining ingredient.

	==Other comments==		==Other comments==

	The definition section may also contain a few other comments, in so far as these comments play a direct role in clarifying the definition. Examples are typically reserved for another section, and under special circumstances, there is a separate section titled ''Importance''.		The definition section may also contain a few other comments, in so far as these comments play a direct role in clarifying the definition. Examples are typically reserved for another section, and under special circumstances, there is a separate section titled ''Importance''.

Vipul: /* Linking style */

2008-05-11T23:17:55Z

Linking style

← Older revision		Revision as of 23:17, 11 May 2008
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	Whenever a term is being defined using other terms, put links to those terms. For instance, when defining a group property, begin with: A [[group]] is said to be ... if ...		Whenever a term is being defined using other terms, put links to those terms. For instance, when defining a group property, begin with: A [[group]] is said to be ... if ...

	Typically, the symbol-free definition will not attempt to define other terms referenced in the definition. The definition with symbols, because it gives a more explicit description, may also give a clearer idea of other terms. For instance, in the symbol-free definition of normality, the ~~first~~ point states that a subgroup is normal if it is invariant under all [[inner automorphism]]s. In the definition with symbols, the explicit form of inner automorphisms is given.		Typically, the symbol-free definition will not attempt to define other terms referenced in the definition. The definition with symbols, because it gives a more explicit description, may also give a clearer idea of other terms. For instance, in the symbol-free definition of normality, the second point states that a subgroup is normal if it is invariant under all [[inner automorphism]]s. In the definition with symbols, the explicit form of inner automorphisms is given.

	===Locating defining ingredients===		===Locating defining ingredients===

Vipul at 23:17, 11 May 2008

2008-05-11T23:17:29Z

Show changes

Vipul: 3 revisions

2008-05-07T22:42:35Z

3 revisions

← Older revision	Revision as of 22:42, 7 May 2008
(No difference)

Vipul at 16:14, 5 July 2007

2007-07-05T16:14:28Z

← Older revision		Revision as of 16:14, 5 July 2007
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			{{manual of style}}

	The '''Definition''' section in Groupprops should serve as a complete introduction to what the term ''means''.		The '''Definition''' section in Groupprops should serve as a complete introduction to what the term ''means''.

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	For most terms, the following two formats are used for specifying the definition:		For most terms, the following two formats are used for specifying the definition:

	* '''Symbol-free definition''': This gives the definition without using any additional symbols. For instance, for permutable subgroup, the symbol-free definition states:		* '''Symbol-free definition''': This gives the definition without using any additional symbols. For instance, for [[permutable subgroup]], the symbol-free definition states:

	A [[subgroup]] of a [[group]] is termed '''permutable''' if its [[product of subgroups\|product]] with every subgroup of the group is a subgroup, or equivalently, if it commutes with every subgroup.		A [[subgroup]] of a [[group]] is termed '''permutable''' if its [[product of subgroups\|product]] with every subgroup of the group is a subgroup, or equivalently, if it commutes with every subgroup.
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	===Equivalence of mutiple definitions===		===Equivalence of mutiple definitions===

	Usually, there is a separate section within the definition part explaining the equivalence of definitions, or giving a link to a page that gives a ''full proof'' of this equivalence. ~~This can be seen, for instance,~~		Usually, there is a separate section within the definition part explaining the equivalence of definitions, or giving a link to a page that gives a ''full proof'' of this equivalence.

	==Links from within the definition==		==Links from within the definition==
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	Typically, the symbol-free definition will not attempt to define other terms referenced in the definition. The definition with symbols, because it gives a more explicit description, may also give a clearer idea of other terms. For instance, in the symbol-free definition of normality, the first point states that a subgroup is normal if it is invariant under all [[inner automorphism]]s. In the definition with symbols, the explicit form of inner automorphisms is given.		Typically, the symbol-free definition will not attempt to define other terms referenced in the definition. The definition with symbols, because it gives a more explicit description, may also give a clearer idea of other terms. For instance, in the symbol-free definition of normality, the first point states that a subgroup is normal if it is invariant under all [[inner automorphism]]s. In the definition with symbols, the explicit form of inner automorphisms is given.

	~~[[Category:Groupprops]]~~

Vipul at 10:27, 15 March 2007

2007-03-15T10:27:07Z

← Older revision		Revision as of 10:27, 15 March 2007
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	==Formats of definition==		==Formats of definition==

			===Symbol-free versus with symbols===

	For most terms, the following two formats are used for specifying the definition:		For most terms, the following two formats are used for specifying the definition:
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	A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''permutable''' if for every subgroup <math>K</math>, <math>HK=KH</math>.		A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''permutable''' if for every subgroup <math>K</math>, <math>HK=KH</math>.


	==Multiple definitions==		==Multiple definitions==

			===Equivalent definitions as a list===

	Often, the same term may have multiple equivalent definitions. In this case, all the equivalent forms are stated as bullet points. The same format is repeated in the symbol-free definition and in the definition with symbols.		Often, the same term may have multiple equivalent definitions. In this case, all the equivalent forms are stated as bullet points. The same format is repeated in the symbol-free definition and in the definition with symbols.
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	* For all <math>g</math> in <math>G</math>, <math>gNg^{-1} = N</math>.		* For all <math>g</math> in <math>G</math>, <math>gNg^{-1} = N</math>.
	* For all <math>g</math> in <math>G</math>, <math>gN = Ng</math>.		* For all <math>g</math> in <math>G</math>, <math>gN = Ng</math>.

			===Separate definitions as separate subsections===

			Sometimes, if the equivalent definitions are long and intricate or both require their own machinery, they may be developed in separate subsections. The title of each subsection gives some ''title'' to the definition.

			For instance, the two definitions of group are labelled as the '''textbook definition''' and the '''universal algebraic definition'''.

			===Equivalence of mutiple definitions===

			Usually, there is a separate section within the definition part explaining the equivalence of definitions, or giving a link to a page that gives a ''full proof'' of this equivalence. This can be seen, for instance,

	==Links from within the definition==		==Links from within the definition==

Vipul at 15:51, 21 February 2007

2007-02-21T15:51:33Z

New page

The '''Definition''' section in Groupprops should serve as a complete introduction to what the term ''means''.

==Formats of definition==

For most terms, the following two formats are used for specifying the definition:

* '''Symbol-free definition''': This gives the definition without using any additional symbols. For instance, for permutable subgroup, the symbol-free definition states:

A [[subgroup]] of a [[group]] is termed '''permutable''' if its [[product of subgroups|product]] with every subgroup of the group is a subgroup, or equivalently, if it commutes with every subgroup.

Here, symbols for the group and subgroup are not introduced.

The advantage of a symbol-free definition is that it sometimes brings out the essence without any symbol cluttering.

For a property operator, a symbol may be used for the property being operated on, to avoid overuse of pronous and to maintain some semblance of clarity.

* '''Definition with symbols''': This gives the definition with explicit introduction of symbols for every quantified object. For instance the definition with symbols of the property of being a [[permutable subgroup]] is:

A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''permutable''' if for every subgroup <math>K</math>, <math>HK=KH</math>.

==Multiple definitions==

Often, the same term may have multiple equivalent definitions. In this case, all the equivalent forms are stated as bullet points. The same format is repeated in the symbol-free definition and in the definition with symbols.

For instance, the symbol-free definition of [[normal subgroup]]:

A [[subgroup]] of a [[group]] is said to be '''normal''' if it satisfies the following equivalent conditions:
* It is invariant under all [[inner automorphism]]s. Thus, normality is the [[invariance property]] with respect to the property of an automorphism being inner.
* It is the kernel of a homomorphism from the group.
* It equals each of its conjugates in the whole group.
* Its left cosets are the same as its right cosets.

The definition with symbols of [[normal subgroup]]:

A subgroup <math>N</math> of a group <math>G</math> is said to be ''normal'' in <math>G</math> (in symbols, <math>N</math> \triangleleft <math>G</math>) if the following equivalent conditions hold:

* For all <math>g</math> in <math>G</math>, <math>gNg^{-1}</math> &sube; <math>N</math>.
* There is a homomorphism <math>\phi</math> from <math>G</math> to another group such that the kernel of <math>\phi</math> is precisely <math>N</math>.
* For all <math>g</math> in <math>G</math>, <math>gNg^{-1} = N</math>.
* For all <math>g</math> in <math>G</math>, <math>gN = Ng</math>.

==Links from within the definition==

Whenever a term is being defined using other terms, put links to those terms. For instance, when defining a group property, begin with: A [[group]] is said to be ... if ...

Typically, the symbol-free definition will not attempt to define other terms referenced in the definition. The definition with symbols, because it gives a more explicit description, may also give a clearer idea of other terms. For instance, in the symbol-free definition of normality, the first point states that a subgroup is normal if it is invariant under all [[inner automorphism]]s. In the definition with symbols, the explicit form of inner automorphisms is given.

[[Category:Groupprops]]