Wikipedia: History of Elementary group theory

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=== First theorems about groups

First Theorems about Groups

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Given a group (G,*) defined as:

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*1). (G,*) has closure. That is, if a and b belong to (G,*), then a*b belongs to (G,*)
*2). The operation * is associative, that is, if a, b, and c belong to (G,*), then (a*b)*c=a*(b*c).
*3). (G,*) contains an identity element, say e, that is, if a belongs to (G,*), then e*a=a*e=a.
*4). Every element in (G,*) has an inverse, that is, if a belongs to (G,*), there is an element b in (G,*) such that a*b=b*a=e.

# (G,*) has closure. That is, if a and b belong to (G,*), then a*b belongs to (G,*)
# The operation * is associative, that is, if a, b, and c belong to (G,*), then (a*b)*c=a*(b*c).
# (G,*) contains an identity element, say e, that is, if a belongs to (G,*), then e*a=a*e=a.
# Every element in (G,*) has an inverse, that is, if a belongs to (G,*), there is an element b in (G,*) such that a*b=b*a=e.

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* The identity element in a GrouP (G,*) is unique.

* The identity element in a group (G,*) is unique.

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* Therefore, the inverse of an element x in a Group, (G,*) is unique.

* Therefore, the inverse of an element x in a group, (G,*) is unique.

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* If this is successful, then the assumption that the proposition is false, is, itself, false. Hence, the proposition is true.

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Four More Elementary Group Theorems

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* The results of Theorem IV are often called the cancellation rules for a Group.

* The results of Theorem IV are often called the cancellation rules for a group.

	Revision 5 . . (edit) September 5, 2001 3:11 am by Zundark [tidy up the format a little]
	Revision 4 . . August 21, 2001 8:23 am by AxelBoldt [some clean up]
	Revision 3 . . August 21, 2001 8:05 am by AxelBoldt [some clean up]