Solvability-forcing number: Difference between revisions

Latest revision as of 14:43, 23 June 2024

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

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[1].

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[2].

@@ Line 8: / Line 8: @@
 * 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==
@@ Line 14: / Line 14: @@
 ===Stronger properties===
-* [[Odd number]]
+{| class="sortable" border="1"
-* [[Nilpotence-forcing number]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Abelianness-forcing number]]
+|-
-* [[Cyclicity-forcing number]]
+| [[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}}
+|-
+| 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::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}}
+|}
+==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].