We will finish with several examples of UnitLib usage to give an idea how to work with the package.
In the first example we retrieve from the library the normalized unit group of the group algebra of the dihedral group of order 128 over the field of two elements, compute its center and find some of its properties, and then check that the group generated by these generators expressed in terms of group algebra elements is abelian:
gap> IdGroup(DihedralGroup(128)); [ 128, 161 ] gap> V := PcNormalizedUnitGroupSmallGroup( 128, 161 ); <pc group of size 170141183460469231731687303715884105728 with 127 generators> gap> C := Center( V ); <pc group with 34 generators> gap> gens := MinimalGeneratingSet( C );; gap> Length(gens); 19 gap> Size(C); 17179869184 gap> AbelianInvariants(C); [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 8, 8, 16, 32 ] gap> KG := UnderlyingGroupRing( V ); <algebra-with-one over GF(2), with 7 generators> gap> f := NaturalBijectionToNormalizedUnitGroup( KG );; gap> IsAbelian( Group( List( gens, x -> x^f ) )); true
In the second example we will check the conjecture about the coincidence of the lower and upper Lie nilpotency indices of the modular group algebras for all non-abelian groups of order 64.
It is known that these indices coincide for p-groups with p>3 [BP92], but in the general case the problem remains open.
The indices t_L(G) and t^L(G) can be computed using the LAGUNA package. While the upper Lie nilpotency index can be expressed only in terms of the underlying group G, the lower Lie nilpotency index is determined by the formula t_L(G) = cl V(KG) + 1 [Du92], and can be computed immediately whenever V(KG) is known.
In the program below we enumerate all groups of size 64 and check the conjecture (we do not exclude from consideration some particular cases when the conjecture is known to be true for p=2, because this is beyond the task of this manual).
gap> for n in [ 1 .. NrSmallGroups( 64 ) ] do > if not IsAbelian( SmallGroup( 64, n ) ) then > V := PcNormalizedUnitGroupSmallGroup( 64, n ); > KG := UnderlyingGroupRing( V ); > if LieLowerNilpotencyIndex( KG ) <> > LieUpperNilpotencyIndex( KG ) then > Print( n," - counterexample !!! \n" ); > break; > fi; > fi; > od;
Thus, the test was finished without finding a counterexample.
In the next example we will answer the question about possible nilpotency classes of normalized unit groups of modular group algebras of nonabelian groups of order 64:
gap> cl := [];; gap> for n in [ 1 .. NrSmallGroups( 64 ) ] do > if not IsAbelian( SmallGroup( 64, n ) ) then > V := PcNormalizedUnitGroupSmallGroup( 64, n ); > AddSet( cl, NilpotencyClassOfGroup( V ) ); > fi; > od; gap> cl; [ 2, 3, 4, 5, 6, 7, 8, 16 ]
With UnitLib you can perform the computation from the last example in several hours on a modern computer. Without UnitLib you will spend the same time to compute only several normalized unit groups V(KG) for groups of order 128 with the help of the LAGUNA package. Note that without LAGUNA such computation would not be feasible at all.
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