Difference between revisions of "Group theory"
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'''Group theory''' is the area of mathematics which deals directly with the study of [[group]]s. | '''Group theory''' is the area of mathematics which deals directly with the study of [[group]]s. | ||
| − | In order for a set | + | In order for a set <math>G</math> to be considered a group, it must have the following four properties: |
| − | 1) An operation * (such as addition or multiplication, although multiplication is standard) is defined on | + | 1) An operation <math>*</math> (such as addition or multiplication, although multiplication is standard) is defined on <math>G</math>. |
| − | 2)G has an has an identity element | + | 2)<math>G</math> has an has an identity element <math>j</math> under <math>*</math> such that for any element <math>a</math> in <math>G</math>, <math>j*a=a*j=a</math>. |
| − | 3)The operation is associative, which means for any three elements | + | 3)The operation is associative, which means for any three elements <math>a</math>, ''b'', and ''c'' in ''G'', (''a''*''b'')*''c''=a*(''b''*''c'') |
4)Every element ''a'' in ''G'' has an inverse "x" under * that is also in ''G'' such that ''a''*''x''=''x''*''a''=''j''. | 4)Every element ''a'' in ''G'' has an inverse "x" under * that is also in ''G'' such that ''a''*''x''=''x''*''a''=''j''. | ||
Revision as of 20:10, 13 March 2008
Group theory is the area of mathematics which deals directly with the study of groups.
In order for a set 
to be considered a group, it must have the following four properties:
1) An operation 
(such as addition or multiplication, although multiplication is standard) is defined on .
2) has an has an identity element 
under such that for any element 
in , 
.
3)The operation is associative, which means for any three elements , b, and c in G, (a*b)*c=a*(b*c)
4)Every element a in G has an inverse "x" under * that is also in G such that a*x=x*a=j.
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