Group

A group is a mathematical structure that abstracts the behavior of symmetries acting on an object. It consists of a set with a given binary operation on it satisfying certain axioms.

Groups are extremely useful throughout mathematics, and are central to the theory of polytopes. Lots of research on polytopes has been devoted to studying their most symmetric classes, including regular and uniform polytopes. The ways in which these are symmetric may be described with groups, including symmetry groups and automorphism groups.

Definition
A group is a set $$G$$ together with a binary operation $$*:G\times G\to G$$ (that is, the operation is closed) such that:


 * For any $$a,b,c\in G$$, $$(a*b)*c=a*(b*c)$$. (Associativity)
 * There exists $$e\in G$$ such that $$a*e=e*a=a$$ for any $$a\in G$$. (Identity element)
 * For any $$a\in G$$ there exists $$a^{-1}\in G$$ such that $$a*a^{-1}=e$$. (Inverse elements)

The identity element $$e$$ may alternatively be notated as $$1$$. If the operation is some form of addition, the identity element may be notated as $$0$$, and the inverse of $$a$$ may be notated as $$-a$$.

Examples
The following are examples of groups.


 * The integers form a group under addition. The identity element is $$0$$ and any number $$a$$ has an inverse $$-a$$.
 * The symmetries of an equilateral triangle form a group under composition, i.e. applying one symmetry after the other. The composition of any two symmetries gives another symmetry. The identity element is the trivial symmetry (the “do-nothing” symmetry), and every symmetry has an inverse symmetry that reverts it (e.g. 120° clockwise rotation is reverted by 120° counterclockwise rotation). This symmetry group is called A2.
 * The automorphisms of a triangle also form a group under composition. As it turns out, this group has exactly the same structure as A2 – they are isomorphic.

Equivalence between groups
Very rarely is exact equality between groups a useful notion. A much more useful notion is that of an isomorphism. Two groups $$(G,\cdot)$$ and $$(H,*)$$ are said to be isomorphic whenever there exists a bijective function $$\varphi:G\to H$$ such that
 * $$\varphi(a\cdot b)=\varphi(a)*\varphi(b)$$ for any $$a,b\in G$$.

However, when dealing with symmetry groups, this can be too loose of a notion. For instance, the symmetry group H3 of the icosahedron and the symmetry group A4+ of the pentachoron are isomorphic, yet clearly deserve to be distinguished. We instead classify symmetry groups up to conjugacy. Two symmetry groups $$G$$ and $$H$$ in the same space $$\mathbb R^n$$ are said to be conjugate whenever there exists some invertible isometry $$f$$ such that
 * $$G=f^{-1}Hf$$.

That is to say, one group is the same as the other "moved around".

Subgroups
A subgroup of $$(G,\cdot)$$ is any group $$(H,\cdot)$$ for $$H\subseteq G$$. Note that $$\cdot$$ must send pairs of elements of $$H$$ to elements of $$H$$ for this to make sense. By Lagrange's theorem, whenever $$G$$ is finite, the order (number of elements) of $$H$$ must divide that of $$G$$.