Solvability-forcing number - Revision history

R-a-jones at 14:43, 23 June 2024

2024-06-23T14:43:55Z

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	\| [[Weaker than::cyclicity-forcing number]] \|\| a number such that any group of that order is cyclic \|\| (via abelianness-forcing) \|\| (via abelianness-forcing) \|\| {{intermediate notions short\|solvability-forcing number\|cyclicity-forcing number}}		\| [[Weaker than::cyclicity-forcing number]] \|\| a number such that any group of that order is cyclic \|\| (via abelianness-forcing) \|\| (via abelianness-forcing) \|\| {{intermediate notions short\|solvability-forcing number\|cyclicity-forcing number}}
	\|}		\|}

			==List==

			The following is a list of all solvability-forcing numbers below 100: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99. That is, every number below 100 except for 60 is solvability-forcing (see [[alternating group:A5]]).
			This sequence is A085736 in the OEIS[https://oeis.org/A085736].

			Perhaps it is more illuminating to list the complement of this sequence: 60, 120, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, ..., which is A056866 in the OEIS[https://oeis.org/A056866].

Vipul: /* Symbol-free definition */

2016-06-20T21:33:22Z

Symbol-free definition

← Older revision		Revision as of 21:33, 20 June 2016
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	* Every group of that order is [[solvable group\|solvable]]		* Every group of that order is [[solvable group\|solvable]]
	* It has no non-prime divisor which is [[simple-feasible number\|simple-feasible]]. In other words, no divisor of it occurs as the order of a simple non-~~Abelian~~ group		* It has no non-prime divisor which is [[simple-feasible number\|simple-feasible]]. In other words, no divisor of it occurs as the order of a simple non-abelian group

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

Vipul: /* Stronger properties */

2016-06-20T21:21:29Z

Stronger properties

← Older revision		Revision as of 21:21, 20 June 2016
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	===Stronger properties===		===Stronger properties===

	* [[Weaker than::~~Odd~~ number]]~~: {{proofat~~\|[[~~Odd~~-order implies solvable]] (also known as the Feit-Thompson theorem or odd-order theorem)}}		{\| class="sortable" border="1"
	* A number whose order has at most two distinct prime factors~~. {{proofat~~\|[[~~Order~~ has only two prime factors implies solvable]] (also known as Burnside's p^aq^b theorem}}		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* A number ~~whose order is the product of~~ three distinct ~~primes. {{proofat\|[[Order is product of~~ three distinct ~~primes implies normal Sylow subgroup]]}}~~		\|-
	* [[Weaker than::~~Square~~-free number]]~~: A~~ number whose order is a product of distinct primes~~. {{proofat~~\|[[Square-free implies solvability-forcing]]}}		\| [[Weaker than::odd number]] \|\| not divisible by 2 \|\| [[odd-order implies solvable]] (also known as the Feit-Thompson theorem or odd-order theorem) \|\| any nontrivial power of 2 offers a counterexample \|\| {{intermediate notions short\|solvability-forcing number\|odd number}}
	* [[Weaker than::~~Nilpotency~~-forcing number]]		\|-
	* [[Weaker than::~~Abelianness~~-forcing number]]		\| a number whose order has at most two distinct prime factors \|\| \|\| [[order has only two prime factors implies solvable]] (also known as Burnside's p^aq^b theorem \|\| any odd number with three or more distinct prime factors; also, any square-free number with three or more distinct prime factors \|\|
	* [[Weaker than::~~Cyclicity~~-forcing number]]		\|-
			\|[[Weaker than::square-free number]]\|\| a number whose order is a product of distinct primes \|\| [[Square-free implies solvability-forcing]] \|\| any [[prime power]], such as the square of a prime \|\| {{intermediate notions short\|solvability-forcing number\|square-free number}}
			\|-
			\| [[Weaker than::nilpotency-forcing number]] \|\| a number such that any group of that order is nilpotent \|\| follows from [[nilpotent implies solvable]] \|\| a number such as 6 is solvability-forcing but not nilpotency-forcing \|\| {{intermediate notions short\|solvability-forcing number\|nilpotency-forcing number}}
			\|-
			\| [[Weaker than::abelianness-forcing number]] \|\| a number such that any group of that order is abelian \|\| (via nilpotency-forcing) \|\| (via nilpotency-forcing) \|\| {{intermediate notions short\|solvability-forcing number\|abelianness-forcing number}}
			\|-
			\| [[Weaker than::cyclicity-forcing number]] \|\| a number such that any group of that order is cyclic \|\| (via abelianness-forcing) \|\| (via abelianness-forcing) \|\| {{intermediate notions short\|solvability-forcing number\|cyclicity-forcing number}}
			\|}

Vipul: /* Relation with other properties */

2016-06-20T21:14:16Z

Relation with other properties

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	* A number whose order is the product of three distinct primes. {{proofat\|[[Order is product of three distinct primes implies normal Sylow subgroup]]}}		* A number whose order is the product of three distinct primes. {{proofat\|[[Order is product of three distinct primes implies normal Sylow subgroup]]}}
	* [[Weaker than::Square-free number]]: A number whose order is a product of distinct primes. {{proofat\|[[Square-free implies solvability-forcing]]}}		* [[Weaker than::Square-free number]]: A number whose order is a product of distinct primes. {{proofat\|[[Square-free implies solvability-forcing]]}}
	* [[Weaker than::~~Nilpotence~~-forcing number]]		* [[Weaker than::Nilpotency-forcing number]]
	* [[Weaker than::Abelianness-forcing number]]		* [[Weaker than::Abelianness-forcing number]]
	* [[Weaker than::Cyclicity-forcing number]]		* [[Weaker than::Cyclicity-forcing number]]

Vipul: /* Relation with other properties */

2009-01-19T23:24:20Z

Relation with other properties

← Older revision		Revision as of 23:24, 19 January 2009
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	===Stronger properties===		===Stronger properties===

	* [[Odd number]]		* [[Weaker than::Odd number]]: {{proofat\|[[Odd-order implies solvable]] (also known as the Feit-Thompson theorem or odd-order theorem)}}
	* [[Nilpotence-forcing number]]		* A number whose order has at most two distinct prime factors. {{proofat\|[[Order has only two prime factors implies solvable]] (also known as Burnside's p^aq^b theorem}}
	* [[Abelianness-forcing number]]		* A number whose order is the product of three distinct primes. {{proofat\|[[Order is product of three distinct primes implies normal Sylow subgroup]]}}
	* [[Cyclicity-forcing number]]		* [[Weaker than::Square-free number]]: A number whose order is a product of distinct primes. {{proofat\|[[Square-free implies solvability-forcing]]}}
			* [[Weaker than::Nilpotence-forcing number]]
			* [[Weaker than::Abelianness-forcing number]]
			* [[Weaker than::Cyclicity-forcing number]]

Vipul: 2 revisions

2008-05-08T00:17:12Z

2 revisions

← Older revision	Revision as of 00:17, 8 May 2008
(No difference)

Vipul at 18:27, 16 June 2007

2007-06-16T18:27:03Z

← Older revision		Revision as of 18:27, 16 June 2007
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	* Every group of that order is [[solvable group\|solvable]]		* Every group of that order is [[solvable group\|solvable]]
	* It has no non-prime divisor which is [[simple-feasible number\|simple-feasible]]. In other words, no divisor of it occurs as the order of a simple non-Abelian group		* It has no non-prime divisor which is [[simple-feasible number\|simple-feasible]]. In other words, no divisor of it occurs as the order of a simple non-Abelian group

			==Relation with other properties==

			===Stronger properties===

			* [[Odd number]]
			* [[Nilpotence-forcing number]]
			* [[Abelianness-forcing number]]
			* [[Cyclicity-forcing number]]

Vipul at 18:25, 16 June 2007

2007-06-16T18:25:09Z

New page

{{natural number property}}

==Definition==

===Symbol-free definition===

A natural number is said to be '''solvability-forcing''' if it satisfies the following equivalent conditions:

* Every group of that order is [[solvable group|solvable]]
* It has no non-prime divisor which is [[simple-feasible number|simple-feasible]]. In other words, no divisor of it occurs as the order of a simple non-Abelian group