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Minimal Non-Solvable Groups
and the Theorem
The minimal finite non-solvable groups
have been classified by Thompson in
1968:
| 1. PSL ( 2 , p ) , | p = 5
or p> 5 prime, p = 2, 3 mod 5 , |
| 2. PSL ( 2 , 2n ) , |
n prime ,
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| 3. PSL ( 2 , 3n
) , | n prime, n>2 , |
| 4. PSL ( 3 , 3 ) , |
| 5. Suzuki ( 2n ) , |
n odd .
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In view of this result Conjecture (1) is implied by
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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
Un(x,y)!=1.
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We prove Conjecture (2):
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Theorem : |
Let w = X-2 Y-1 X
and let G
be one of the groups 1.- 5. of Thompsons's list.
Then, neither of the identities Un (x,y)
= 1 holds everywhere in G.
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--> sufficient: show that there exist x
, y in G such that
1 != U1 (x,y) = U2 (x,y) .
--> we compute
U2 = [X-1 Y-1, Y
X-2 Y-1 X Y-1] =
X-3 Y-1 X2 Y
X-1 Y X2 Y-1
Proof with
Computer and Algebraic Geometry.
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