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<title>GAP (UnitLib) - Chapter 4: An example of UnitLib usage</title>
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<h3>4. An example of <strong class="pkg">UnitLib</strong> usage</h3>
<p>We will finish with several examples of <strong class="pkg">UnitLib</strong> usage to give an idea how to work with the package.</p>
<p>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 express one of its generators in terms of group algebra elements:</p>
<table class="example">
<tr><td><pre>
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> KG := UnderlyingGroupRing( V );
<algebra-with-one over GF(2), with 7 generators>
gap> f := NaturalBijectionToNormalizedUnitGroup( KG );;
gap> gens[8]^f;
(Z(2)^0)*f3+(Z(2)^0)*f4+(Z(2)^0)*f7+(Z(2)^0)*f3*f4+(Z(2)^
0)*f3*f5+(Z(2)^0)*f3*f6+(Z(2)^0)*f3*f7+(Z(2)^0)*f4*f5+(Z(2)^
0)*f4*f6+(Z(2)^0)*f4*f7+(Z(2)^0)*f3*f4*f5+(Z(2)^0)*f3*f4*f6+(
Z(2)^0)*f3*f4*f7+(Z(2)^0)*f3*f5*f6+(Z(2)^0)*f3*f5*f7+(Z(2)^
0)*f3*f6*f7+(Z(2)^0)*f4*f5*f6+(Z(2)^0)*f4*f5*f7+(Z(2)^
0)*f4*f6*f7+(Z(2)^0)*f3*f4*f5*f6+(Z(2)^0)*f3*f4*f5*f7+(Z(2)^
0)*f3*f4*f6*f7+(Z(2)^0)*f3*f5*f6*f7+(Z(2)^0)*f4*f5*f6*f7+(Z(2)^
0)*f3*f4*f5*f6*f7
</pre></td></tr></table>
<p>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.</p>
<p>It is known that these indices coincide for p-groups with p>3 <a href="chapBib.html#biBBhandari-Passi">[BP92]</a>, but in the general case the problem remains open.</p>
<p>The indices t_L(G) and t^L(G) can be computed using the <strong class="pkg">LAGUNA</strong> 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 <a href="chapBib.html#biBDu">[D92]</a>, and can be computed immediately whenever V(KG) is known.</p>
<p>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).</p>
<table class="example">
<tr><td><pre>
gap> for n in [ 1 .. NrSmallGroups( 64 ) ] do
> if not IsAbelian( SmallGroup( 64, n ) ) then
> Print( n, "\r" );
> V := PcNormalizedUnitGroupSmallGroup( 64, n );
> KG := UnderlyingGroupRing( V );
> if LieLowerNilpotencyIndex( KG ) <>
> LieUpperNilpotencyIndex( KG ) then
> Print( n," - counterexample !!! \n" );
> break;
> fi;
> fi;
> od;
gap>
</pre></td></tr></table>
<p>Thus, the test was finished without finding a counterexample.</p>
<p>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 128:</p>
<table class="example">
<tr><td><pre>
gap> cl := [];
gap> for n in [ 1 .. NrSmallGroups( 128 ) ] do
> if not IsAbelian( SmallGroup( 128, n ) ) then
> Print( n, "\r" );
> V := PcNormalizedUnitGroupSmallGroup( 128, n );
> AddSet( cl, NilpotencyClassOfGroup( V ) );
> fi;
> od;
gap> cl;
[ 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 32 ]
</pre></td></tr></table>
<p>With <strong class="pkg">UnitLib</strong> you can perform the computation from the last example in several hours on a modern computer. Without <strong class="pkg">UnitLib</strong> 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 <strong class="pkg">LAGUNA</strong> package. Note that without <strong class="pkg">LAGUNA</strong> such computation would not be feasible at all.</p>
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