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Minimal Non-Solvable Groups and the Theorem
The minimal finite non-solvable groups have been classified by Thompson in 1968:
 1.  PSL ( 2 , p ) , p = 5   or  p > 5 prime,  p = + - 2 mod 5 2.  PSL ( 2 , 2n ) , n prime 3.  PSL ( 2 , 3n ) , n prime,   n>2 4.  PSL ( 3 , 3 ) , 5.  Suzuki ( 2n ) . n   odd
In view of this result Conjecture (1) is implied by
 Conjecture (2): There exists a word w in  X, X-1, Y, Y-1,  such that for all G in the above list and for all n there exist x, y in G such that Un ( x , y )  !=   1.
We prove Conjecture (2) for all but one cases:
 Theorem : Let  w   =   X-1 Y X Y-1 X   and let G be one of the groups 1.- 4. of Thompsons's list. Then, neither of the identities   Un ( x , y )   =   1   holds everywhere in G.
--> sufficient: show that there exist x , y in G such that 1 != U1 ( x , y ) = U2 ( x , y ) . Proof with Computer and Algebraic Geometry.

Lille, 08-07-02 http://www.singular.uni-kl.de