group Group 2001-08-29 18:17:07 2007-06-18 19:41:01 Definition identity inverse neutralizing element non-trivial element nontrivial element group operation ring algebra morphism subgroup group set \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} \textbf{Group.}\\ A group is a pair $(G,\,*)$, where $G$ is a non-empty set and ``$*$'' is a binary operation on $G$, such that the following conditions hold: \begin{itemize} \item For any $a,b$ in $G$, \,$a*b$\, belongs to $G$. (The operation ``$*$'' is closed). \item For any \,$a,b,c\in G$, \,$(a*b)*c=a*(b*c)$. \,(Associativity of the operation). \item There is an element $e\in G$ such that \,$g*e=e*g=g$\, for any \,$g\in G$. (Existence of identity element). \item For any \,$g\in G$\, there exists an element $h$ such that \,$g*h=h*g=e$. \,(Existence of inverses). \end{itemize} If $G$ is a group under *, then * is referred to as the \emph{group operation} of $G$. Usually, the symbol ``$*$'' is omitted and we write \,$ab$\, for $a*b$. \,Sometimes, the symbol ``$+$'' is used to represent the operation, especially when the group is \emph{abelian}. It can be proved that there is only one identity element, and that for every element there is only one inverse. \,Because of this we usually denote the inverse of $a$ as $a^{-1}$ or $-a$ when we are using additive notation. \,The identity element is also called \emph{neutral element} due to its behavior with respect to the operation, and thus $a^{-1}$ is sometimes (although uncommonly) called the {\em neutralizing element} of $a$. An element of a group besides the identity element is sometimes called a \emph{non-trivial element}. Groups often arise as the symmetry groups of other mathematical objects; the study of such situations uses group actions. \,In fact, much of the study of groups themselves is conducted using group actions.