Finite simple group - Revision history

Vipul: /* Stronger properties */

2011-08-01T23:19:38Z

Stronger properties

← Older revision		Revision as of 23:19, 1 August 2011
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	===Stronger properties===		===Stronger properties===

	* [[Finite simple non-~~Abelian~~ group]]		* [[Finite simple non-abelian group]]

	===Weaker properties===		===Weaker properties===

Vipul at 21:23, 1 July 2008

2008-07-01T21:23:59Z

← Older revision		Revision as of 21:23, 1 July 2008
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	{{group property conjunction\|[[finite group]]\|[[simple group]]}}		{{group property conjunction\|finite group\|simple group}}

	{{semibasicdef}}		{{semibasicdef}}

Vipul: 3 revisions

2008-05-07T23:29:55Z

3 revisions

← Older revision	Revision as of 23:29, 7 May 2008
(No difference)

Vipul: /* Relation with other properties */

2007-03-25T01:51:43Z

Relation with other properties

← Older revision		Revision as of 01:51, 25 March 2007
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	* [[Simple group]]		* [[Simple group]]
			* [[K-group]]

	==Related notions==		==Related notions==

Vipul: /* Feasibility */

2007-03-25T01:51:26Z

Feasibility

← Older revision		Revision as of 01:51, 25 March 2007
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	To answer this question, we need a mix of techniques from Sylow theory, representation theory, group actions, and of course basic arguments with subgroups.		To answer this question, we need a mix of techniques from Sylow theory, representation theory, group actions, and of course basic arguments with subgroups.

			{{further\|[[simple-feasible number]]}}

Vipul at 01:50, 25 March 2007

2007-03-25T01:50:00Z

New page

{{group property conjunction|[[finite group]]|[[simple group]]}}

{{semibasicdef}}

{{cfsg}}

==Definition==

===Symbol-free definition===

A [[group]] is said to be a '''finite simple group''' if it satisfies both these conditions:

* It is a [[finite group]], viz its [[order]] (the cardinality of its underlying set) is a finite integer
* It is a [[simple group]], viz it has no proper nontrivial [[normal subgroup]]

==Relation with other properties==

===Stronger properties===

* [[Finite simple non-Abelian group]]

===Weaker properties===

* [[Simple group]]

==Related notions==

===Classification problem===

The Classification problem is the problem of obtaining a complete description or classification of all the finite simple groups. The Classification Problem has now been solved, viz it has been shown that every finite simple group is one of these:

* A [[cyclic group]] of prime order
* An [[alternating group]]
* A [[classical group]] viz a group obtained as a subquotient of one of the typical linear groups (special linear group, orthogonal group, unitary group, symplectic group)
* An [[exceptional group of Lie type]]
* A [[sporadic simple group]]

{{further|[[Classification of finite simple groups]]}}

===Feasibility===

A natural number is said to be [[simple-feasible number|simple-feasible]] if it occurs as the order of a simple group. One of the important initial questions in the classification of simple groups is: what are the simple-feasible numbers?

To answer this question, we need a mix of techniques from Sylow theory, representation theory, group actions, and of course basic arguments with subgroups.