
Minimal NonSolvable Groups
and the Theorem
The minimal finite nonsolvable 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 , 2^{n} ) , 
n prime

3. PSL ( 2 , 3^{n}
) ,  n prime, n>2 
4. PSL ( 3 , 3 ) , 
5. Suzuki ( 2^{n} ) . 
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
U_{n} ( 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 U_{n} ( x , y )
= 1 holds everywhere in G.

> sufficient: show that there exist x
, y in G such that
1 != U_{1} ( x , y ) = U_{2} ( x , y
) .
Proof with
Computer and Algebraic Geometry.
