Group and abelian group

A group is a monoid $(G,*,e)$ (Semigroup and monoid) in which every element is invertible (Left inverse, right inverse, and invertible element of a monoid). Written out, a group is a set $G$ with a binary operation $*$ (Binary operation on a set; associativity, commutativity, and a subset closed under the operation) and an element $e \in G$ such that

• (G1) $(x * y) * z = x * (y * z)$ for all $x, y, z \in G$;
• (G2) $e * x = x = x * e$ for every $x \in G$;
• (G3) for every $x \in G$ there is $y \in G$ with $y * x = e = x * y$.

By A left identity and a right identity for the same binary operation are equal; hence there is at most one two-sided identity the element $e$ of (G2) is unique, and by In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided the element $y$ of (G3) is unique for each $x$ and is determined by either one of the two equations alone. So the notations $e$ for the identity and $x^{-1}$ for the inverse of $x$ are legitimate, and asserting $y * x = e$ alone already says $y = x^{-1}$.

A group is abelian, or commutative, when its operation is commutative: $x * y = y * x$ for all $x, y \in G$.

Notation. The operation is usually written multiplicatively, $xy$ for $x * y$, with identity $e$ (or $1$) and inverse $x^{-1}$. For an abelian group it is often written additively, $x + y$, with identity $0$ and inverse $-x$; the two notations describe the same notion and nothing below depends on the choice. We write $G$ for the group when the operation is understood.

• Why (G3) may be stated one-sidedly in practice. The axiom as written asks for a two-sided inverse. Because In a monoid, a left inverse and a right inverse of the same element are equal; hence an invertible element has exactly one inverse, and it is two-sided is proved before this definition, and is proved in the sharp form "a left inverse and a right inverse of the same element coincide", verifying either equation for a candidate $y$ suffices once the element is known to be invertible. Nothing on this page ever uses a one-sided inverse without that lemma standing behind it.

• A group is a monoid, not merely like one. Every statement proved for monoids applies verbatim to groups. In the other direction the invertible elements of a monoid form a group (The invertible elements of a monoid form a group under the restricted operation), so groups arise inside every monoid.

• The axioms are not independent of the notation, but the content is. Read additively, (G1)–(G3) are exactly the additive axioms in the definition of a field (Field), whose axiom (A) says in as many words that $(F,+)$ is an abelian group and whose axiom (M) says the same of $(F \setminus \{0\}, \cdot)$; so every field already carries two groups.