Group

We say that (G, *) is a group if it is a Monoid that also satisfies the following property:

• Invertibility: For every x ∈ G there is an xinv such that x * xinv = xinv * x = e

Moreover, it is an abelian group if it satisfies the property:

• Commutative: x * y = y * x for all x and y ∈ G.

Examples of Groups

• Integers Z are an abelian group under addition
• Natural numbers are not a group under addition (given a number x in N, -x is not in N)
• Neither integers nor natural numbers are a group under multiplication, but the set of nonzero rational numbers (n/d for any n, d ∈ N, n ≠ 0, d ≠ 0) is an (abelian) group under multiplication.

Documentation Help

We’d love your help fleshing out this documentation! You can edit this page in your browser by clicking this link. These links might be helpful:

• Monoid.scala