Group
A group is an algebraic structure which is a monoid with the additional requirement of a unary operation, the inverse, which gives rise to the identity element.
Written out explicitly, a group is a set \(G\) with a binary operation \(\ast\) satisfying the following properties:
- Closure: \(\forall a, b \in G\), \(a \ast b \in G\).
- Associativity: \(\forall a, b, c \in G\), \(a \ast (b \ast c) = (a \ast b) \ast c\).
- Existence of an identity: \(\exists e \in G, \forall a \in G\): \(a \ast e = e \ast a = a\). This element \(e\) is called the identity.
- Existence of an inverse: \(\forall a \in G, \exists a' \in G: a \ast a' = a' \ast a = e\). This element \(a'\) is called the inverse.
Notes on Notation
In groups where the binary operation is multiplication, often \(a^{-1}\) is the notation used for the inverse, whereas where the underlying operation is addition, \(-a\) will often be used.
The same is true for the use of \(1\) and \(0\) for the multiplicative and additive identity respectively, while \(\mathrm{id}\) is often used in a more general context.
Similarly \(na\) may be used to denote \(a \ast a \ast a \dots a\) with \(n\) terms if the underlying operation is addition, or \(a^{n}\) if the underlying operation is multiplication, despite the fact that in the first case multiplication need not be defined and in the second case exponentiation need not be defined.