(T. Bandman, G.-M. Greuel, F. Grunewald, B. Kunyavskii, G. Pfister, E. Plotkin)
| Problem: Characterize the class of finite solvable groups by 2-variable identities. |
| Examples: | * | 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. |
| * |
A finite group G is nilpotent
<==> [[...[[x,y],y]...],y] = 1
for all x,y in G (some n-fold commutator: Engel identity). |
For any word w in X, Y, X-1, Y-1 consider the sequence (Un) of words (depending on w)
| U1 | = | w |
| Un+1 | = | [ X UnX-1 , Y UnY-1 ] |
| Conjecture (1): | (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
Un ( x , y ) = 1
for all x,
y in G
|