Solvability-forcing number

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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, Square-free number|FULL LIST, MORE INFO