# Definitions > [!NOTE] **Definition** (Group Axioms) > A *group* is an [[Algebraic Structure|algebraic structure]] $(G,*),$ where $G$ is a set and $*$ is a [[Binary Operation|binary operation]], that satisfies: > > ($G0$) Closure: $\forall a,b\in G\;(a*b \in G)$ (even though $*$ by definition is closed on its domain this axiom is still needed in the case that the domain of $*$ is a superset of $G$) > > ($G1$) Associativity: $*$ is [[Associativity|associative]] on $G$ (we may write $xy$ instead $x*y$) > > ($G2$) [[Group Identity Element|Identity]]: The [[Identity element of a binary operation|identity]] of $*$ lies in $G$ (usually denoted $e,$ $1$ or $0$) > > ($G3$) [[Inverse of Group Element|Inverse]]: Every member $x$ of $G$ has an [[Inverse under a binary operation|inverse]] under $*$ in $G$ (usually denoted $x^{-1}$) > > [!Example] > The set of symmetric transformations of a square of a group form a group, under composition, known as the [[Dihedral Group D8|dihedral group of order eight]] $D_{8}.$ # Properties A group is said to [[Abelian Group|abelian]] if its binary operation is additionally *commutative.* **Consequences of group axioms**: The group axioms assert that the [[Uniqueness of Group Identity|identity element is unique]] and that each element has a [[Uniqueness of Group Inverses|unique inverse]]. The 'Socks-Shoes property' is that [[Inverse of the Product of Group Elements|inverse of the product]] of group elements is given by $(ab)^{-1}=b^{-1}a^{-1}.$ **Exponentiation**: Let $n\in\mathbb{Z}.$ The $n$th [[Integer Power of Group Element|power]] of a group element $a$ is defined recursively as $a^{n}=1,$ if $n=0,$ and $a^{n}=a^{n-1}a$ for $n>0.$ If $n<0,$ then $a^{n}=(a^{-n})^{-1}.$ The [[Order of Group Element|order of a group element]] $a$ is defined as the smallest natural number $n$ that satisfies $a^{n}=1.$ If no such positive integer exists $a$ is said to have infinite order. **Size**: The group $(G,*)$ is said to be a [[Finite Group|finite group]] if $G$ is a [[Finite Set|finite set]]. The order of a finite group is defined as the cardinality of its set. Note that [[Element of a finite group is of finite order|every element of a finite group has finite order]]. Also, a consequence of *Lagrange's theorem* is that the [[Order of Element of Finite Group Divides Order of The Group|order of any element of a finite group divides the order of the group]]. **Subgroups & Quotients**: $(H,*)$ is a [[Subgroup|subgroup]] of $(G,*)$ if it satisfies the group axioms and $H\subset G.$ Let $g \in G.$ The [[Generated Subgroup|subgroup generated]] by $g$ is the set containing all powers $g.$ A group is [[Cyclic Group|cyclic]] if it is equal to the subgroup by one of its members. [[Lagrange's theorem (on Finite Groups)|Lagrange's theorem]] asserts that the order of a subgroup of a finite group divides the order of the finite group. [[Coset]] . A subgroup is [[Normal Subgroup|normal]] iff the quotient is a group. **Comparing Groups**: A [[Homomorphism of Groups|homomorphism]] between groups is a mapping that preserves the group operation. An [[Isomorphism of Groups|isomorphism]] between groups is a bijective homomorphism between them. The [[Direct Product of Groups|direct product]] of groups is the cartesian product of their sets under the operation $(g_{1},h_{1})\circ(g_{2},h_{2})=(g_{1}\circ_{1}g_{2}, h_{1} \circ_{2}h_{2}).$ # Applications **Examples**: (Infinite Groups) [[Integers]]; [[Euclidean Group]]; [[Orthogonal Group]]; [[Special Orthogonal Group]]; (Finite Groups) [[Dihedral Group]]; [[Symmetric Groups of Finite Degree]]; [[nth Alternating Group]]; **Generalisations**: