group

Group

Group

Mathematically a Group is a Monoid G all of whose elements have multiplicative inverses.

Caution

No assumptions are made whether or not the group is Abelian.

Important

Contract: Group initializer parameters must have

• mult closed and associative on reps

• one an identity on reps, rep*one == rep == one*rep

• inv must me idempotent: inv(inv(rep)) == rep

class boring_math.abstract_algebra.algebras.group.Group

Bases: Monoid, Generic

Parameters:

• mult – Associative function H X H -> H on representations.

• one – Representation for multiplicative identity.

• invert – Function H -> H mapping element representation to the representation of corresponding inverse element.

• narrow – Narrow the rep type, many-to-one function. Like choosing an element from a coset of a group.

__init__(mult: Callable[[H, H], H], one: H, invert: Callable[[H], H], narrow: Callable[[H], H] | None = None)

Parameters:

• mult – Associative function H X H -> H on representations.

• one – Representation for multiplicative identity.

• invert – Function H -> H mapping element representation to the representation of corresponding inverse element.

• narrow – Narrow the rep type, many-to-one function. Like choosing an element from a coset of a group.

class boring_math.abstract_algebra.algebras.group.GroupElement

Bases: MonoidElement, Generic

__init__(rep: H, algebra: Group[H]) → None

__str__() → str

Returns:

str(self) = GroupElement<rep>

__pow__(n: int) → Self

Raise the element to the power of n.

Parameters:

n – The int power to raise the element to.

Returns:

The element (or its inverse) raised to the integer``n`` power.

Raises:

• ValueError – If element’s algebra

• ValueError – If self and other are same type but different concrete groups.

• ValueError – If algebra fails to have an identity or elements not invertible.

__mul__(right: object) → Self

Multiply two elements of the same concrete algebra together.

Parameters:

right – An element within the same concrete algebra or a right action.

Returns:

The product self * right otherwise NotImplemented.

Raises:

ValueError – If self and right are same type but different concrete algebras.

__rmul__(left: object) → Self

When left side of multiplication does not know how to multiply right side.

Parameters:

left – Left side of the multiplication.

Returns:

Never returns, otherwise left.__mul__(right) would have worked.

Raises:

TypeError – When multiplying on left by an int.