[GAP Forum] Constructing a 2-Engel group

[Werner Nickel]

Dear Dr Abdollahi, dear GapForum,

> Does there exist a finite 2-Engel 3-group G such that
> |G|=3^{11},
> G^3=G'=C_9 X C_9 X C_9 X C_3,
> Z_2(G)=C_9 X C_9 X C_9 X C_9,
>  \Omega_1(G)=Z(G) \cap G'=(Z_2(G))^3 and exp(G)=27,
> where \Omega_1(G) is the subgroup of G generated by all
> elements x in G such that x^3=1.
> 
> Thanks in advance for any help.

I cannot construct a group with all the properties specified above.
However, it is an easy calculation in GAP (with the packages
polycyclic and nq installed) to construct 2-Engel groups with
exponent 27.

>From the conditions on the order of the group and the derived group,
your group is a 4-generators group.  The following computation in
GAP gives the 4-generators 3-Engel group of Exponent 27:

LoadPackage( "nq" );
F := FreeGroup("a","b","c","d", "x","y","z");
A := F / [ LeftNormedComm( [F.5,F.6,F.6] ), F.7^27 ];
G := NilpotentQuotient( A, [F.5,F.6,F.7] );

The generators  F.5,F.6,F.7  are used as identical generators.
Therefore, the two relations specify the 2-Engel law and the exponent
law.

This produces a group of size 3^34.  Its commutator quotient is C_27^4
instead of the required C_3^4.  My attempts to force the commutator
factor group to have the right form (by adding relations of the type
g^3 = element of G') always produced groups with elementary abelian
derived group.

Perhaps someone else has an idea how one can explore the quotients of
the group above in a systematic way to prove or disprove that a group
with the required properties exists.

All the best,
Werner Nickel

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   Dr (AUS) Werner Nickel         Mathematics with Computer Science
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