A group is a set with a distinguished element e and a binary operation [ , ] satisfying: ‣ [a,b] = [[a,c],[b,c]] ‣ [a,e] = a ‣ [a,a] = e for all a,b,c.
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@johncarlosbaez can mention of the identity element be dropped with these three equations? [a,b]=[[a,c],[b,c]] [a,[b,b]]=a [a,a]=[b,b] What do we get with just [a,b]=[[a,c],[b,c]] [a,[b,b]]=a ?
Yes, you can drop mention of the identity as mentioned. In group theory there's a famous easy proof that if ax = a for all a and ya = a for all a then x = y, so if there are right units and left units they must all be equal.
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Derp. I knew that.
thanks for the kind reminder.Hvala. Twitter će to iskoristiti za poboljšanje vaše vremenske crte. PoništiPoništi
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What do we gain or lose by mentioning a special element in a definition? It would appear the operation IS the group, and that a definition mentioning an element weakens this tool of definition
Hvala. Twitter će to iskoristiti za poboljšanje vaše vremenske crte. PoništiPoništi
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