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Definition (Group)
Let $G$ be a nonempty set, and let $*$ be a binary operation on $G$, which means for every $x,y\in G$ we have $x*y\in G$ $\textbf{[Closure]}$. The structure $\mathbb{G} = (G,*)$ is called a $\textdef{group}$ if the following hold.
- $\textbf{[Associativity]}$ For all $x,y,z\in G$ we have $(x* y)* z = x* (y* z)$.
- $\textbf{[Identity]}$ There is an $e\in G$ such that for all $x\in G$ we have $x * e = e* x = x$.
- $\textbf{[Inverses]}$ For all $x\in G$ there is a $y\in G$ such that $x* y = y* x = e$.
We usually simply write $G$ when referring to the entire structure $\mathbb{G}=(G,*)$. The element $e$ from the second point is called the $\textdef{identity}$. The element $y$ from the third point is called the $\textdef{inverse}$ of $x$ and is usually denoted $x^{-1}$. One often simply writes $xy$ in place of $x*y$, and for every positive integer $n$, we'll write $x^n$ as shorthand for $x* x* \cdots * x$ ($n$ times).
The following pages link to this page.
- Definition.Group
- Definition.GroupFooter
- Definition.GroupHomomorphism
- Definition.GroupIsomorphism
- Definition.OrderForGroups
- Definition.Subgroup
- Definition.TheSubgroupGeneratedByASubset
- Problem.CanWeUseDivisionToCreateAGroup
- Problem.SubgroupsAreSubsetsThatAreClosedUnderProductsAndInverse
- Problem.ZNAndUNAreGroups
- Schedule.20131011
- Schedule.20131016
- Schedule.20131018
- Schedule.20131028
- Schedule.20131104
- Schedule.20131106
- Schedule.20161007
- Schedule.20161010
- Schedule.20161012
- Schedule.20161014
- Schedule.20161017
- Schedule.20161031
- Schedule.20161107
- Schedule.20161109
- Schedule.20161111
- Schedule.20171013
- Schedule.20171016
- Schedule.20171020
- Schedule.20171106
- Schedule.20171108
- Schedule.20171110
- Schedule.AllProblems
- Schedule.ExamStuff