Finite solvable group - Revision history

Vipul: /* Stronger properties */

2015-12-26T02:45:14Z

Stronger properties

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	\|-		\|-
	\| [[Weaker than::finite supersolvable group]] \|\| finite and a [[supersolvable group]] \|\| follows from [[supersolvable implies solvable]] \|\| see [[solvable not implies supersolvable]] \|\| {{intermediate notions short\|finite solvable group\|finite supersolvable group}}		\| [[Weaker than::finite supersolvable group]] \|\| finite and a [[supersolvable group]] \|\| follows from [[supersolvable implies solvable]] \|\| see [[solvable not implies supersolvable]] \|\| {{intermediate notions short\|finite solvable group\|finite supersolvable group}}
			\|-
			\| [[Weaker than::group having subgroups of all orders dividing the group order]] \|\| finite and has subgroups of all orders dividing the group order \|\| [[subgroups of all orders dividing the group order implies solvable]] \|\| \|\| {{intermediate notions short\|finite solvable group\|group having subgroups of all orders dividing the group order}}
	\|}		\|}

Vipul: /* Examples */

2015-12-26T01:12:08Z

Examples

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	* [[symmetric group:S5]], [[special linear group:SL(2,5)]], [[direct product of A5 and Z2]]: All of these are groups of order 120 which have [[alternating group:A5]] as one of their [[composition factor]]s, and are hence neither simple nor solvable.		* [[symmetric group:S5]], [[special linear group:SL(2,5)]], [[direct product of A5 and Z2]]: All of these are groups of order 120 which have [[alternating group:A5]] as one of their [[composition factor]]s, and are hence neither simple nor solvable.
	* [[projective special linear group:PSL(3,2)]]: This is a simple non-abelian group of order 168.		* [[projective special linear group:PSL(3,2)]]: This is a simple non-abelian group of order 168.

	~~===Groups satisfying the property===~~

	~~{{groups satisfying property by importance rank}}~~

	~~===Groups dissatisfying the property===~~

	~~{{groups dissatisfying property sorted by importance rank}}~~

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Examples */

2015-12-26T01:11:51Z

Examples

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	* [[symmetric group:S5]], [[special linear group:SL(2,5)]], [[direct product of A5 and Z2]]: All of these are groups of order 120 which have [[alternating group:A5]] as one of their [[composition factor]]s, and are hence neither simple nor solvable.		* [[symmetric group:S5]], [[special linear group:SL(2,5)]], [[direct product of A5 and Z2]]: All of these are groups of order 120 which have [[alternating group:A5]] as one of their [[composition factor]]s, and are hence neither simple nor solvable.
	* [[projective special linear group:PSL(3,2)]]: This is a simple non-abelian group of order 168.		* [[projective special linear group:PSL(3,2)]]: This is a simple non-abelian group of order 168.

			===Groups satisfying the property===

			{{groups satisfying property by importance rank}}

			===Groups dissatisfying the property===

			{{groups dissatisfying property sorted by importance rank}}

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Examples based on order */

2015-12-26T01:02:36Z

Examples based on order

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	* Any product of two prime powers, i.e., any number of the form <math>p^aq^b</math>, with <math>p,q</math> primes, is solvability-forcing. See [[order has only two prime factors implies solvable]] (this result is also termed Burnside's <math>p^aq^b</math>-theorem).		* Any product of two prime powers, i.e., any number of the form <math>p^aq^b</math>, with <math>p,q</math> primes, is solvability-forcing. See [[order has only two prime factors implies solvable]] (this result is also termed Burnside's <math>p^aq^b</math>-theorem).
	* Any odd number is solvability-forcing. See [[odd-order implies solvable]]. This result, also called the '''odd-order theorem''' or the Feit-Thompson theorem, is highly nontrivial.		* Any odd number is solvability-forcing. See [[odd-order implies solvable]]. This result, also called the '''odd-order theorem''' or the Feit-Thompson theorem, is highly nontrivial.
	* Any square-free number i.e., any number that is a product		* Any [[square-free number]] i.e., any number that is a product of pairwise distinct primes. See [[square-free implies solvability-forcing]].

	===Non-examples===		===Non-examples===

Vipul: /* Relation with other properties */

2012-06-06T03:10:04Z

Relation with other properties

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	\| [[polycyclic group]] \|\| has a [[subnormal series]] (of finite length) with cyclic quotient groups \|\| stronger than solvability \|\| {{intermediate notions short\|solvable group\|polycyclic group}}		\| [[polycyclic group]] \|\| has a [[subnormal series]] (of finite length) with cyclic quotient groups \|\| stronger than solvability \|\| {{intermediate notions short\|solvable group\|polycyclic group}}
	\|-		\|-
	\| [[locally solvable group]] \|\| every [[finitely generated group\|finitely generated]] subgroup is solvable \|\| weaker than solvability \| {{intermediate notions short\|locally solvable group\|solvable group}}		\| [[locally solvable group]] \|\| every [[finitely generated group\|finitely generated]] subgroup is solvable \|\| weaker than solvability \|\| {{intermediate notions short\|locally solvable group\|solvable group}}
	\|-		\|-
	\| [[hypoabelian group]] \|\| the transfinite [[derived series]] reaches the identity element \|\| \|\| weaker than solvability \|\| {{intermediate notions short\|hypoabelian group\|solvable group}}		\| [[hypoabelian group]] \|\| the transfinite [[derived series]] reaches the identity element \|\| weaker than solvability \|\| {{intermediate notions short\|hypoabelian group\|solvable group}}
	\|-		\|-
	\| [[residually solvable group]] \|\| every non-identity element is outside a [[normal subgroup]] for which the [[quotient group]] is solvable \|\| weaker than solvability \|\| {{intermediate notions short\|residually solvable group\|solvable group}}		\| [[residually solvable group]] \|\| every non-identity element is outside a [[normal subgroup]] for which the [[quotient group]] is solvable \|\| weaker than solvability \|\| {{intermediate notions short\|residually solvable group\|solvable group}}

Vipul at 23:37, 22 March 2012

2012-03-22T23:37:14Z

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	==Relation with other properties==		==Relation with other properties==

			===Properties whose conjunction with finiteness gives this property===

			Below is a list of [[group property\|group properties]] such that a [[finite group]] has the property if and only if it is a [[finite solvable group]].

			{\| class="sortable" border="1"
			! Property !! Meaning !! Relation with solvability in general !! Intermediate properties between it and solvability in general
			\|-
			\| [[solvable group]] \|\| [[derived series]] reaches identity in finitely many steps \|\| same \|\| --
			\|-
			\| [[polycyclic group]] \|\| has a [[subnormal series]] (of finite length) with cyclic quotient groups \|\| stronger than solvability \|\| {{intermediate notions short\|solvable group\|polycyclic group}}
			\|-
			\| [[locally solvable group]] \|\| every [[finitely generated group\|finitely generated]] subgroup is solvable \|\| weaker than solvability \| {{intermediate notions short\|locally solvable group\|solvable group}}
			\|-
			\| [[hypoabelian group]] \|\| the transfinite [[derived series]] reaches the identity element \|\| \|\| weaker than solvability \|\| {{intermediate notions short\|hypoabelian group\|solvable group}}
			\|-
			\| [[residually solvable group]] \|\| every non-identity element is outside a [[normal subgroup]] for which the [[quotient group]] is solvable \|\| weaker than solvability \|\| {{intermediate notions short\|residually solvable group\|solvable group}}
			\|}

	===Stronger properties===		===Stronger properties===

Vipul: /* Non-examples */

2012-01-22T21:44:52Z

Non-examples

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	* [[alternating group:A5]]: This is the smallest order simple non-abelian group. It has order 60. See [[A5 is the simple non-abelian group of smallest order]]		* [[alternating group:A5]]: This is the smallest order simple non-abelian group. It has order 60. See [[A5 is the simple non-abelian group of smallest order]]
	* [[symmetric group:S5]], [[special linear group:SL(2,5)]], [[direct product of A5 and Z2]]: All of these are groups of order 120 which have [[alternating group:A5]] as one of their [[composition factor]]s, and are hence ~~not~~ simple.		* [[symmetric group:S5]], [[special linear group:SL(2,5)]], [[direct product of A5 and Z2]]: All of these are groups of order 120 which have [[alternating group:A5]] as one of their [[composition factor]]s, and are hence neither simple nor solvable.
	* [[projective special linear group:PSL(3,2)]]: This is a simple non-abelian group of order 168.		* [[projective special linear group:PSL(3,2)]]: This is a simple non-abelian group of order 168.

Vipul at 03:16, 17 December 2011

2011-12-17T03:16:22Z

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	{{finite group property}}		{{finite group property}}
			[[importance rank::2\| ]]
	==Definition==		==Definition==

Vipul: /* Definition */

2011-09-11T17:07:36Z

Definition

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	# It has [[Sylow complement]]s for all prime divisors of the order of the group		# It has [[Sylow complement]]s for all prime divisors of the order of the group
	# It has [[Hall subgroup]]s of all possible orders		# It has [[Hall subgroup]]s of all possible orders
	# All its composition ~~factors~~ (i.e., the quotient groups for any [[defining ingredient::composition series]] for the group) are cyclic groups of prime order.		# All its [[composition factor]]s (i.e., the quotient groups for any [[defining ingredient::composition series]] for the group) are cyclic groups of prime order. Equivalently, all its composition factors are abelian.
			# All its [[chief factor]]s (i.e., the successive quotient groups for any [[defining ingredient::chief series]] for the group) are [[elementary abelian group]]s.

	===Equivalence of definitions===		===Equivalence of definitions===

Vipul: /* Examples */

2011-07-07T16:00:15Z

Examples

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	* The [[trivial group]] is a finite solvable group.		* The [[trivial group]] is a finite solvable group.

			===Examples based on order===

			We call a natural number <math>n</math> a [[solvability-forcing number]] if every group of order <math>n</math> is solvable. It turns out that:

			* Any [[prime power]] is solvability-forcing, because [[prime power order implies nilpotent]] and [[nilpotent implies solvable]].
			* Any product of two prime powers, i.e., any number of the form <math>p^aq^b</math>, with <math>p,q</math> primes, is solvability-forcing. See [[order has only two prime factors implies solvable]] (this result is also termed Burnside's <math>p^aq^b</math>-theorem).
			* Any odd number is solvability-forcing. See [[odd-order implies solvable]]. This result, also called the '''odd-order theorem''' or the Feit-Thompson theorem, is highly nontrivial.
			* Any square-free number i.e., any number that is a product

			===Non-examples===

			Any finite simple non-abelian group is a finite group that is not solvable. See [[classification of finite simple groups]] for a list of finite simple non-abelian groups.

			Further, any group that contains a finite simple non-abelian group as a [[subgroup]], has a finite simple non-abelian group as a [[quotient group]], or admits a finite simple non-abelian group as a [[subquotient]] must be non-solvable.

			The smallest order examples of finite non-solvable groups are below:

			* [[alternating group:A5]]: This is the smallest order simple non-abelian group. It has order 60. See [[A5 is the simple non-abelian group of smallest order]]
			* [[symmetric group:S5]], [[special linear group:SL(2,5)]], [[direct product of A5 and Z2]]: All of these are groups of order 120 which have [[alternating group:A5]] as one of their [[composition factor]]s, and are hence not simple.
			* [[projective special linear group:PSL(3,2)]]: This is a simple non-abelian group of order 168.

	==Relation with other properties==		==Relation with other properties==