I would suggest that the proof of Burnside using group characters that a group of order $p^{a}q^{b}$ ($p$,$q$ primes) is solvable is beautiful, but not really explanatory, from a group theoretical point of view. The same can be probably be said for Frobenius's characterization of finite permutation groups in which no non-identity element fixes more than one point. While there are now "explanatory" proofs of the first result using purely group-theoretic methods,(which are arguably less beautiful) there is presently no purely group-theoretic proof of the second (although Terry Tao has recently found an alternative proof using character theory of certain commutative algebras, which can be seen on his blog).

I would suggest that the proof of Burnside using group characters that a group of order $p^{a}q^{b}$ ($p$,$q$ primes) is solvable is beautiful, but not really explanatory, from a group theoretical point of view. The same can be probably be said for Frobenius's characterization of finite permutation groups in which no non-identity element fixes more than one point. While there are now "explanatory" proofs of the first result using purely group-theoretic methods,(which are arguably less beautiful) there is presently no purely group-theoretic proof of the second (although Terry Tao has recently found an alternative proof using character theory of certain commutative algebras, which can be seen on his blog).

I would suggest that the proof of Burnside using group characters that a group of order $p^{a}q^{b}$ ($p$,$q$ primes) is solvable is beautiful, but not really explanatory, from a group theoretical point of view. The same can be probably be said for Frobenius's characterization of finite permutation groups in which no non-identity element fixes more than one point. While there are now ``explanatory" proofs of the first result using purely group-theoretic methods,(which aree arguably less beautiful) there is presently no purely group-theoretic proof of the second (although Terry Tao has recently found an alternative proof using character theory of certain commutative algebras, which can be seen on his blog).

I would suggest that the proof of Burnside using group characters that a group of order $p^{a}q^{b}$ ($p$,$q$ primes) is solvable is beautiful, but not really explanatory, from a group theoretical point of view. The same can be probably be said for Frobenius's characterization of finite permutation groups in which no non-identity element fixes more than one point. While there are now "explanatory" proofs of the first result using purely group-theoretic methods,(which are arguably less beautiful) there is presently no purely group-theoretic proof of the second (although Terry Tao has recently found an alternative proof using character theory of certain commutative algebras, which can be seen on his blog).