Order-forcing operator

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This article describes a property operator that takes as input a group property and outputs a property of natural numbers

Definition

The order-forcing operator is a property operator from the group property space to the natural number property space that takes as input a given group property p and outputs the property of being a natural number such that every group with that natural number as order must satisfy property p.

Examples

Cyclicity

Further information: cyclicity-forcing number

Applying the order-forcing operator to the group property of being cyclic gives the natural number property of being cyclicity-forcing. A natural number is said to be cyclicity-forcing if every group of that order is cyclic.

Any prime number, and further any square-free number subject to certain additional congruence conditions, is cyclicity-forcing.

Abelianness

Further information: Abelianness-forcing number

Applying the order-forcing operator to the group property of being Abelian gives the natural number property of being Abelianness-forcing. A natural number is said to be Abelianness-forcing if every group with that as order, must be Abelian.

Nilpotence

Further information: Nilpotence-forcing number

Applying the order-forcing operator to the group property of being nilpotent gives the natural number property of being nilpotence-forcing. A natural number is said to be nilpotence-forcing if every group with that as order, must be nilpotent.

For instance, any prime power is nilpotence-forcing.

Solvability

Further information: Solvability-forcing number

Applying the order-forcing operator to the group property of being solvable gives the natural number property of being solvability-forcing. A natural number is said to be solvability-forcing if every group with that as order, must be solvable.

By the Feit-Thompson theorem, any odd number is solvability-forcing. By Burnside's p^aq^b theorem, any number with only two prime factors is solvability-forcing.