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Algorithm to sovle the problem: G is solvable


Strictly speaking, this should not be called an "algorithm". It works only if you are asked, in a qual or some other test to prove that a group is solvable.


The ultimate goal is to find a normal subgroup. Then we are left with two smaller groups and they usually make it kind of obvious or easy enough to see that the normal subgroup and the factor groups are solvable. Then by extension theorem, done! (Or if you meet someone really mean, you can always recursively apply this "algorithm". )


Note: You need to have a couple of typical solvable groups in your head. Small groups are solvable. (Everything is true with small numbers. That's how mathematicians were at all able to find out some truth.). P-groups are solvable (Why? They have a non-trivial center and just use induction.) I think that is about it.


First, apply Sylow theorems to your given group. To all its p-groups where p is a prime factor of the order. So, you have possibly a11, or a12... p1 p-Sylow subroups, a21, a22... p2 p-Sylow subgroups. If one of them is 1, boom, you just found yourself a normal subgroup. How easy is that!


Second, if that doesn't work out, you look at one of the p-Sylow with the smallest index. Where are we going? Yeah, we use representation on cosets. If the order of the group doesn't divide the index's factorial, then by representation on cosets, boom, you found a normal subgroup.


Third, if they both don't work out which is seldom the case unless you are left with some really small groups. You can see if you can "count" the elements and get a contradiction.


If none of the above gives you a normal group and you are as assumed given the problem: prove, a group with n elements is solvable, then, go over it again. I mean what else do you know or within test time can give you a proof that a group is solvable. (Don't tell me a group is solvable if and only if it's the product of its p-Sylows. If you were able to prove it's a product of p-Sylows, you would have got some result already at 1.) So, that's the "algorithm". 25% on your qual in 1 min. Enjoy.





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