
Equations in Finite Linear Groups
(F. Grunewald, B. Kunyavskii, E. Plotkin and Y. Segev)
Problem:
Characterize the class of finite solvable groups by 2variable identities.

Example: 
A group G is abelian <==> [x,y] = 1
for all x,y in G
where [X,Y] = X Y X^{1} Y^{1}
is the commutator.

For any word w in X,
Y, X^{1}, Y^{1}
consider the sequence (U_{n}) of words
(depending on w)
U_{1} 
= 
w 
U_{n+1} 
= 
[ X U_{n}X^{1} ,
Y U_{n}Y^{1} ] 
Conjecture: 
(B. Plotkin)
There exists a word w such that a
finite group G is solvable if
and only if there is a positive n such that
U_{n} ( x , y ) = 1
for all x,
y in G

Minimal NonSolvable Groups
and the Theorem
