# Elementary group theory

 related topics {math, number, function} {group, member, jewish}

In mathematics and abstract algebra, a group is the algebraic structure $\{G,\perp\}$, where G is a non-empty set and $\perp$ denotes a binary operation $\perp:G\times{}G\rightarrow{}G,$ called the group operation. The notation $\perp(x,y)$ is normally shortened to the infix notation $x\perp{}y$, or even to xy.

A group must obey the following rules (or axioms). Let a,b,c be arbitrary elements of G. Then:

• A1, Closure. $a\perp{}b\in{}G$. This axiom is often omitted because a binary operation is closed by definition.
• A2, Associativity. $(a\perp{}b)\perp{}c=a\perp(b\perp{}c)$.
• A3, Identity. There exists an identity (or neutral) element $e\in{}G$ such that $a\perp{}e=e\perp{}a=a$. The identity of G is unique by Theorem 1.4 below.
• A4, Inverse. For each $a\in{}G$, there exists an inverse element $x\in{}G$ such that $a\perp{}x=x\perp{}a=e$. The inverse of a is unique by Theorem 1.5 below.

An abelian group also obeys the additional rule:

• A5, Commutativity. $a\perp{}b=b\perp{}a$.