Solvability-forcing number: Difference between revisions

Revision as of 21:21, 20 June 2016

This article defines a property that can be evaluated for natural numbers

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
It has no non-prime divisor which is 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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
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	\|FULL LIST, MORE INFO
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
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	\|FULL LIST, MORE INFO
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	\|FULL LIST, MORE INFO
abelianness-forcing number	a number such that any group of that order is abelian	(via nilpotency-forcing)	(via nilpotency-forcing)	\|FULL LIST, MORE INFO
cyclicity-forcing number	a number such that any group of that order is cyclic	(via abelianness-forcing)	(via abelianness-forcing)	Abelianness-forcing number\|FULL LIST, MORE INFO

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 ===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}}
+|}