Group can be determined by how elements multiply with each other

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tgt
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Given a group, it seems everything (all properties) about the group can be determined by how elements multiply with each other. Correct? For example, everything about the group could be read off from the multiplication table.
 

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1. Closure. For all a, b in G, the result of the operation ab is also in G.
2. Associativity. For all a, b and c in G, the equation (ab)c = a(bc) holds.
3. Identity element. There exists an element 1 in G, such that for all elements a in G, the equation 1a = a1 = a holds.
4. Inverse element. For each a in G, there exists an element b in G such that ab = ba = 1, where 1 is the identity element.

After looking at the axioms one at a time, it sure looks like they all can be satisfied for a finite group in a multiplication table.
 
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mathwonk
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well it is hard to look at the multiplication table of a group of order 60 and decide whether it is simple or not, but in principle yes, everything is determined by the multiplication table.

but thats like saying everything about a solid is determined by its equation, but you still may have trouble computing its volume from that knowledge. e.g. the length of the graph of y = x^3 is determined by that equation, but what is that length from x=0 to x=1?
 
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but still, even if you don't have a group of a that big order, say you have a group of 6 elements, then it would be a pain to check the associativity from the operation table. since there would be x nr of cases, where x is the nr of permutations with repitition.
 
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HallsofIvy
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Given a group, it seems everything (all properties) about the group can be determined by how elements multiply with each other. Correct? For example, everything about the group could be read off from the multiplication table.
Well, yeah! A group is defined by its members and its operation- there is nothing else. Of course that operation is not always defined in terms of a multiplication table, but you could, theoretically, write one. (That might be very difficult in the case of infinite groups!)
 

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