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(F. Grunewald, B. Kunyavskii, E. Plotkin and Y. Segev)
| Problem: Characterize the class of finite solvable groups by 2-variable 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 (Un) of words (depending on w)
| U1 | = | w |
| Un+1 | = | [ X UnX-1 , Y UnY-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
Un ( x , y ) = 1
for all x,
y in G
|