Group

Traits

• Group: a generic group $(G,\cdot )$ where $G$ is a set and $\cdot$ is a binary operation.
• FiniteGroup: a group with finite elements defined using Finite trait
• AbelianGroup: group with commutative operation
• FiniteCyclicGroup: a finite group with a generator.

Group

Group represents a group with finite elements. It defines a binary operation on the set of elements of the group.

• IDENTITY: The identity element of the group.
• inverse(&self) -> Self: inverse of the element.
• operation(a: &Self, b: &Self) -> Self: the operation of the group.
• scalar_mul(&self, scalar: &Self::Scalar): multiplication of the element by a scalar.

Structs

The structs that implement these traits are

• MultiplicativePrimeGroup

MultiplicativePrimeGroup

The MultiplicativePrimeGroup struct is a wrapper around a usize that defines $(Z/nZ{)}^{\ast }$ for a prime power $n=p^k$ with binary operation as $\times$:

#![allow(unused)]
fn main() {
pub struct MultiplicativePrimeGroup<const P: usize, const K: usize>(usize);
}

It uses compile time assertions to check that $P$ is prime.

Examples

Symmetric Group example showcases how Group trait is implemented for any struct.