Just to address the theoretical question of computing centric subgroups:
If P is a proper centric subgroup of S, then it is contained in some
maximal subgroup M of S and so C_S(M) <= C_S(P) <= P <= M.
A simple algorithmic change then is to use Centralizer( S,
FrattiniSubgroup( S ) ) instead of Center( S ). This improved
performance a fair amount in both GAP and magma. For GAP, 2^6 went from
30 seconds to 7 seconds, and 2^7 went from 500 seconds to 80 seconds.
Of course, you could also check which maximal subgroups of S are
centric, and only consider their subgroups, but it looked like over 80%
of maximal subgroups were centric, and nearly 80% of groups had all
maximals centric, so it may not be worth the headache.
Kasper Andersen wrote:
Hi,
In connection with a joint project with Bob Oliver and Joana Ventura I
have been carrying out a number of computations for the 2-groups of
order at most 2^8 in Magma. To check them and to learn GAP at the same
time, I have tried to port my programs to GAP.
Unfortunately GAP seems to be significantly slower (a factor of acround
2.7) than Magma. Before I go any further I wonder if there is a way to
speed up GAP. As a test I have been running the following GAP program:
iscritical := function(S,P)
return 1;
end;
criticalsubgroups := function(S)
local kappa,SmodZ,L;
if IsAbelian(S) then
return [];
else
# Only check subgroups containg Z(S)
kappa:=NaturalHomomorphismByNormalSubgroup(S,Center(S));
SmodZ:=FactorGroup(S,Center(S));
L:=SubgroupsSolvableGroup(SmodZ);
L:=List(L,x->PreImage(kappa,x));
return Filtered(L,P->(iscritical(S,P) <> -1));
fi;
end;
n:=2^7;
for i in [1..NumberSmallGroups(n)] do
S:=SmallGroup(n,i);
L:=criticalsubgroups(S);
Print(n," ",i,"\n");
od;
Some comments are in order: To simplify the timings I have left out the
code for the function iscritical, so the above program simply computes
the conjugacy classes of subgroups P containing Z(S) for all 2-groups S
of order 2^7. Moreover my original function does not return true/false
but rather -1 (=false), 0 (=undecided) and 1 (=true). For simplicity I
have left this in.
Unfortunaly even this simple program runs a factor of 2.7 slower than
the corresponding Magma program. Are there any ways to speed this up?
Another question: One of the conditions for criticality is that P is
centric in S, i.e. that C_S(P) is contained in P. Does anyone know any
better algorithm for computing all such subgroups than testing all
subgroups containing Z(S) one by one?
best wishes,
Kasper Andersen
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