Minimal Non-Solvable Groups
and the Theorem
The minimal finite non-solvable groups
have been classified by Thompson in
In view of this result Conjecture (1) is implied by
| 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
We prove Conjecture (2) for all but one cases:
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.
--> sufficient: show that there exist x
, y in G such that
1 != U1 ( x , y ) = U2 ( x , y
Computer and Algebraic Geometry.
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.