Difference between revisions of "Group (Mathematics)"

Revision as of 22:56, November 24, 2009

In modern algebra, a group is a set of objects, together with a rule of composition, which obey certain rules.

A group is a set of objects, sometimes referred to as $G\$ , and a rule of composition, $*\$ , such that

There is an element of $G\$ , call it $e\$ , such that for any $x\$ in $G\$ , $e*x=x*e=x\$ . This element is called the identity of the group. We write $x\in G\$ to mean that $x\$ is in $G\$ .
For any two $x,y\in G\$ , their composition $x*y\in G\$ .
For any three $x,y,z\in G,x*(y*z)=(x*y)*z\$

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 A group is a set of objects, sometimes referred to as <math>G \ </math>, and a rule of composition, <math>* \ </math>, such that
-#. There is an element of <math>G \ </math>, call it <math>e \ </math>, such that for any <math>x \ </math> in <math>G \ </math>, <math>e*x = x*e = x \ </math>.  This element is called the identity of the group.  We write <math>x \in G \ </math> to mean that <math>x \ </math> is in <math>G \ </math>.
+#There is an element of <math>G \ </math>, call it <math>e \ </math>, such that for any <math>x \ </math> in <math>G \ </math>, <math>e*x = x*e = x \ </math>.  This element is called the identity of the group.  We write <math>x \in G \ </math> to mean that <math>x \ </math> is in <math>G \ </math>.
+#For any two <math>x, y \in G \ </math>, their composition <math>x*y \in G \ </math>.
-#. For any two <math>x, y \in G \ </math>, their composition <math>x*y \in G \ </math>.
+#For any three <math>x,y,z \in G, x*(y*z) = (x*y)*z \ </math>
-#. For any three <math>x,y,z \in G, x*(y*z) = (x*y)*z \ </math>
 [[Category:Algebra Terms]]