Difference between revisions of "Group"

Revision as of 15:51, 16 October 2006

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A group $G$ is a set of elements together with an operation $\cdot:G\times G\to G$ (the dot is frequently supressed) satisfying the following conditions:

For all $a,b,c\in G$ , $a(bc)=(ab)c$ (associativity).
There exists an element $e\in G$ so that for all $g\in G$ , $ge=eg=g$ (identity).
For any $g\in G$ , there exists $g^{-1}\in G$ so that $gg^{-1}=g^{-1}g=e$ ( inverses).

One can also thing of groups as monoids with inverses.

Note that the group operation need not be commutative. If the group operation is commutative, we call the group an abelian group (after the Norwegian mathematician Niels Henrik Abel).

Groups frequently arise as permutations of collections of objects. For example, the rigid motions of $\mathbb{R}^2$ that fix a certain regular $n$ -gon is a group, called the dihedral group and denoted $D_{2n}$ (since it has $2n$ elements). Another example of a group is the symmetric group $S_n$ of all permutations of $\{1,2,\ldots,n\}$ .

Related algebraic structures are rings and fields.

@@ Line 7: / Line 7: @@
 * There exists an element <math>e\in G</math> so that for all <math>g\in G</math>, <math>ge=eg=g</math> ([[identity]]).
 * For any <math>g\in G</math>, there exists <math>g^{-1}\in G</math> so that <math>gg^{-1}=g^{-1}g=e</math> ([[Inverse with respect to an operation | inverses]]).
+One can also thing of groups as [[monoid]]s with inverses.
 Note that the group operation need not be [[commutative]].  If the group operation is commutative, we call the group an [[abelian group]] (after the Norwegian mathematician Niels Henrik Abel).

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Difference between revisions of "Group"

Revision as of 15:51, 16 October 2006