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- Thread starter tgt
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- #2

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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

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- #3

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.

- #5

HallsofIvy

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Well, yeah! A group is

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