Dear Siddhartha Sarkar:
I'm not sure exactly what you would like to know about the group
SmallGroup(81,10), but I'll mention one property that is not well
known.
This has to do with groups having "hidden" primes, a term first used
in the paper:
@article {MR0291272,
AUTHOR = {Kov\'{a}cs, L. G. and Neub\"{u}ser, Joachim and Neumann, B. H.},
TITLE = {On finite groups with ``hidden'' primes},
JOURNAL = {J. Austral. Math. Soc.},
FJOURNAL = {Australian Mathematical Society. Journal. Series A. Pure
Mathematics and Statistics},
VOLUME = {12},
YEAR = {1971},
PAGES = {287--300},
ISSN = {0263-6115},
MRCLASS = {20D99},
MRNUMBER = {0291272},
MRREVIEWER = {F. Levin},
}
Let me begin by giving the smallest interesting example of a group
with hidden primes
G = SmallGroup(72,41) = (Z_3 x Z_3) semi Q_8
where Q_8 acts via its embedding in GL(2,F_3). In fact, the Q_8 acts
fix point freely on Z_3 x Z_3; This is of course an example of a Frobenius
group with Q_8 as the Frobenius complement. This is a rank 2 example.
The generalization is to complements where not every element acts fix point
freely, as for Q_8, but only the elements of the "complement" which
are in a minimum generating set (size 2 here) are required to act fix
point freely.
For our example above d(Q_8) = d(G) = 2. G has 1728 generating sets of
size 2; all are pairs of elements of order 4.
The "generating exponent" for a group H, denoted genexp(H), is the least
common multiple of the orders of the elements in minimum generating sets,
of size d(H). In our example, genexp(Q_8) = genexp(G) = 4, while
exp(G) = 12. The prime 3 is a hidden prime.
Non abelian Frobenius complements have been classified, and there
exist those which are p groups only for p = 2.
However, for the property where only the elements of the group which
are elements of minimum generating sets, size d(H), are required to
act fix point freely (I call this "generator fix point free", gfpf),
there exist non-abelian p-groups of all possible ranks which have this
property:
@article {MR0384926,
AUTHOR = {Wall, G. E.},
TITLE = {Secretive prime-power groups of large rank},
JOURNAL = {Bull. Austral. Math. Soc.},
FJOURNAL = {Bulletin of the Australian Mathematical Society},
VOLUME = {12},
YEAR = {1975},
NUMBER = {3},
PAGES = {363--369},
ISSN = {0004-9727},
MRCLASS = {20D15},
MRNUMBER = {0384926},
MRREVIEWER = {Thomas Laffey},
DOI = {10.1017/S000497270002400X},
URL = {https://doi.org/10.1017/S000497270002400X},
}
An undergraduate/graduate student and I (Paul K. Young, 2005),
determined that for rank 2, there exists a unique smallest such
p-group, that group has order p^(1+p), as well as a number of
special properties.
I'll denote this group by T_p = T_{p,2}.
The smallest examples of these groups are
T_2 = SmallGroup(8,4) = Q_8
T_3 = SmallGroup(81,10)
T_5 = SmallGroup(15625,635)
As I understand it, the gap forum does not allow attachments, so
I'll send you separately pictures (via Xgap) of
the lattice of all subgroups of SmallGroup(81,10)
(this is rather cluttered)
81.10.pdf
and the lattice of normal subgroups of SmallGroup(15625,635)
(this is rather striking)
15625.635.normal.pdf
Note that in both cases the normal subgroups are marked in green,
they form a line of length p-1, and the smallest nontrivial one
is the center = Agemo of order p. That is true in general.
d(T_p) = 2 and all minimum generators have order p^2;
their squares generate the cyclic center.
In fact, one can use gap to construct these unique groups:
Tp:=function(p)
local A,a,C,c,W,A0,A1,A2,Tp;
# R. K. Dennis, Cornell, 2007
A:=CyclicGroup(IsPermGroup,p);
C:=CyclicGroup(IsPermGroup,p^2);
a:=GeneratorsOfGroup(A)[1];
c:=GeneratorsOfGroup(C)[1];
W:=WreathProduct(C,A);
A0:=Product(List([1..p],i->Image(Embedding(W,i),c)));
A1:=A0*Image(Embedding(W,p+1),a);
A2:=A0^(p-1)*Image(Embedding(W,p),c^p);
Tp:=Subgroup(W,[A1,A2]);
return Tp;
end;
I'd like to thank Bettina Eick who wrote some programs to help me
generate examples to study at a crucial point. It might be that
similar ideas (using ANUPQ) might help to study the higher rank cases:
For larger ranks r > 2, the only thing I know for sure, is that there
is a unique one for p = 2:
T_{2,3} = SmallGroup(128,802)
It has order 128 = 2^(1+2+4).
One might guess an analogous result:
There is a unique smallest such non-abelian p-group, for all primes p,
and ranks r > 1, and that it has order p^(1+p+p^2+...+p^(r-1)).
Of course, the only evidence I have for this is what's given above.
The structure of T_{2,3} seems to be a bit more complicated than
that of the rank 2 case.
I should mention that an invariant, which I call the
{\it Scharlau Invariant}, first defined in
@article {MR0322035,
AUTHOR = {Scharlau, Winfried},
TITLE = {Eine {I}nvariante endlicher {G}ruppen},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {130},
YEAR = {1973},
PAGES = {291--296},
ISSN = {0025-5874},
MRCLASS = {20C05 (12A50)},
MRNUMBER = {0322035},
MRREVIEWER = {W. E. Jenner},
DOI = {10.1007/BF01246626},
URL = {https://doi.org/10.1007/BF01246626},
}
was determined to always be a prime (Young, senior thesis, Cornell,
2004) (Dennis and Young, preprint) was generalized for the gfpf
problem (the generalized invariant can be any non-negative integer);
all minimum generating sets for a finite group are used to define a
certain ideal in the integral group ring which plays exactly the same
role as the "Scharlau ideal" played in the Scharlau invariant problem.
Ok, this is all rather brief, perhaps even too long.
I can add more if wanted.
Best regards,
Keith Dennis
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