An multiplicative group is a Group whose operation can be thought of as multiplication in some sense.

For example, the multiplicative group of rationals \( (ℚ, 1, *) \).

Infix alias for multiplicative inverse.

Examples:

>>> let x = Product (4 :: Rational)
>>> x / 2
Product {getProduct = 2 % 1}

Infix alias for multiplicative (<>).

Examples:

>>> Product (2 :: Rational) * Product (3 :: Rational)
Product {getProduct = 6 % 1}

Infix alias for power.

Examples:

>>> let x = Product (3 :: Rational)
>>> x ^ 3
Product {getProduct = 27 % 1}

Multiply an element of a multiplicative group by itself n-many times.

This represents ℕ-indexed powers of an element g of a multiplicative group, i.e. iterated products of group elements. This is representable by the universal property \( C(x, ∏_n g) ≅ C(x, g)^n \).

Examples:

>>> power (Product (3 :: Rational)) 3
Product {getProduct = 27 % 1}

A multiplicative abelian group is a Group whose operation can be thought of as commutative multiplication in some sense. Almost all multiplicative groups are abelian.