Solvability-forcing number
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].