Historical definitions of group - Revision history

Vipul at 23:52, 16 September 2009

2009-09-16T23:52:21Z

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			{{historical definitions\|group}}

	Refer [[history of groups]] for more about the history of the definition and concept of group, and [[funky definitions of group]] for a list of weird definitions of group.		Refer [[history of groups]] for more about the history of the definition and concept of group, and [[funky definitions of group]] for a list of weird definitions of group.

Vipul: /* References */

2008-09-04T22:54:05Z

References

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	* {{paperlink\|Cayley54}}		* {{paperlink\|Cayley54}}
			* {{paperlink\|JordanonGalois69}}

	===Textbook references===		===Textbook references===

	* {{booklink\|JordanGT}}		* {{booklink\|JordanGT}}

Vipul: /* Definitions using permutations */

2008-09-04T22:53:06Z

Definitions using permutations

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	==Definitions using permutations==		==Definitions using permutations==

	===Jordan ~~(1870)~~===		===Jordan===

			In a paper in 1869 titled ''Commentaire sur Galois'', Jordan defines a group as follows:

			{{quotation\|A system of substitutions forms a group if the product of any two substitutions of the system is again a member of the system.}}

	In his book ''Traité des substitutions'', Jordan defines a permutation (he calls it a substitution) and then goes on to define a group:		In his book ''Traité des substitutions'', Jordan defines a permutation (he calls it a substitution) and then goes on to define a group:

Vipul at 21:17, 26 August 2008

2008-08-26T21:17:55Z

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	==References==		==References==

			===Paper references===

	* {{paperlink\|Cayley54}}		* {{paperlink\|Cayley54}}

			===Textbook references===

			* {{booklink\|JordanGT}}

Vipul at 21:17, 26 August 2008

2008-08-26T21:17:27Z

← Older revision		Revision as of 21:17, 26 August 2008
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	Refer [[history of groups]] for more about the history of the definition and concept of group, and [[funky definitions of group]] for a list of weird definitions of group.		Refer [[history of groups]] for more about the history of the definition and concept of group, and [[funky definitions of group]] for a list of weird definitions of group.

	==Cayley (1854)==		==Definitions using permutations==

			===Jordan (1870)===

			In his book ''Traité des substitutions'', Jordan defines a permutation (he calls it a substitution) and then goes on to define a group:

			{{quotation\|We will say that a system of substitutions form a group if the product of any two substitutions in the system is itself in the system.}}

			==Definitions in the abstract setting==
			===Cayley (1854)===

	Cayley attempted the first general definition of a group, in his paper ''On the theory of groups, as depending on the symbolic equation <math>\theta^n = 1</math>''. This definition is not precisely the same as the modern definition, though it is equivalent. Cayley states:		Cayley attempted the first general definition of a group, in his paper ''On the theory of groups, as depending on the symbolic equation <math>\theta^n = 1</math>''. This definition is not precisely the same as the modern definition, though it is equivalent. Cayley states:
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	This definition is today summarized as saying that a group is an associative quasigroup. It is equivalent to the modern definition of group. {{further\|[[Associative quasigroup equals group]]}}		This definition is today summarized as saying that a group is an associative quasigroup. It is equivalent to the modern definition of group. {{further\|[[Associative quasigroup equals group]]}}

	==Weber (1882)==		===Weber (1882)===

	Heinrich Weber gave this definition in a paper on quadratic forms.		Heinrich Weber gave this definition in a paper on quadratic forms.

Vipul at 20:58, 26 August 2008

2008-08-26T20:58:08Z

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	Weber's definition is equivalent to the modern definition of [[finite group]].		Weber's definition is equivalent to the modern definition of [[finite group]].

			==References==

			* {{paperlink\|Cayley54}}

Vipul at 20:52, 26 August 2008

2008-08-26T20:52:29Z

← Older revision		Revision as of 20:52, 26 August 2008
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			Refer [[history of groups]] for more about the history of the definition and concept of group, and [[funky definitions of group]] for a list of weird definitions of group.

	==Cayley (1854)==		==Cayley (1854)==

Vipul: New page: ==Cayley (1854)== Cayley attempted the first general definition of a group, in his paper ''On the theory of groups, as depending on the symbolic equation $\theta^n = 1$ ''. This...

2008-08-26T20:51:46Z

New page: ==Cayley (1854)== Cayley attempted the first general definition of a group, in his paper ''On the theory of groups, as depending on the symbolic equation <math>\theta^n = 1</math>''. This...

New page

==Cayley (1854)==

Cayley attempted the first general definition of a group, in his paper ''On the theory of groups, as depending on the symbolic equation <math>\theta^n = 1</math>''. This definition is not precisely the same as the modern definition, though it is equivalent. Cayley states:

{{quotation|A set of symbols <math>1, \alpha, \beta, \dots</math>, all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a ''group''. These symbols are not in general convertible (commutative) but associative, it follows that if the entire group is multiplied by any one of the symbols, either as further or nearer factor (left or right), the effect is simply to reproduce the group''.}}

This definition is today summarized as saying that a group is an associative quasigroup. It is equivalent to the modern definition of group. {{further|[[Associative quasigroup equals group]]}}

==Weber (1882)==

Heinrich Weber gave this definition in a paper on quadratic forms.

{{quotation|A system <math>G</math> of <math>h</math> arbitrary elements <math>\theta_1, \theta_2, \dots, \theta_h</math> is called a group of degree <math>h</math> if it satisfies the following conditions:<br>
I. By some rule which is designated as composition or multiplication, from any two elements of the same system one derives a new element of the same system.<br>
II. It is always true that <math>(\theta_r \theta_s)\theta_t = \theta_r(\theta_s\theta_t)</math>.<br>
III. From <math>\theta \theta_r = \theta \theta_s</math> it follows that <math>\theta_r = \theta_s</math>.}}

Weber's definition is equivalent to the modern definition of [[finite group]].

Historical definitions of group - Revision history

Vipul at 23:52, 16 September 2009

Vipul: /* References */

Vipul: /* Definitions using permutations */

Vipul at 21:17, 26 August 2008

Vipul at 21:17, 26 August 2008

Vipul at 20:58, 26 August 2008

Vipul at 20:52, 26 August 2008

Vipul: New page: ==Cayley (1854)== Cayley attempted the first general definition of a group, in his paper ''On the theory of groups, as depending on the symbolic equation \theta^n = 1''. This...

Vipul: New page: ==Cayley (1854)== Cayley attempted the first general definition of a group, in his paper ''On the theory of groups, as depending on the symbolic equation $\theta^n = 1$ ''. This...