[GAP Forum] The Higman-Thompson group
Stefan Kohl
kohl at mathematik.uni-stuttgart.de
Thu Jul 17 15:45:46 BST 2008
Dear Forum,
Two weeks ago, I have posted the following example of a finitely-generated
infinite simple group to the group-pub-forum:
Def.: Given disjoint residue classes r_1(m_1) and r_2(m_2) of the
integers, let the class transposition (r_1(m_1),r_2(m_2)) be the
permutation which interchanges r_1 + k * m_1 and r_2 + k * m_2
for each integer k and which fixes all other points.
Then our group is
G := < (0(2),1(4)), (0(4),1(4)), (1(4),2(4)), (2(4),3(4)) >.
Having loaded the RCWA package, this group can be entered into GAP by
gap> G := Group(List([[0,2,1,4],[0,4,1,4],[1,4,2,4],[2,4,3,4]],
> ClassTransposition));
<rcwa group over Z with 4 generators>
Last week, in an answer to my posting John P. McDermott reported that he
has found out that this group is isomorphic to the (first) Higman-Thompson
group, which is defined and investigated in
[Higman74] Graham Higman. Finitely Presented Infinite Simple Groups.
Notes on Pure Mathematics, 1974, Department of Pure Mathematics,
Australian National University, Canberra, ISBN 0 7081 0300 6.
The 'standard generators' kappa, lambda, mu and nu given there correspond
to (0(2),1(2)), (1(2),2(4)), (0(2),1(4)) and (1(4),2(4)), respectively.
As the Higman-Thompson group is simple, verifying the isomorphism
requires now only a (very quick and easy) computational check whether
the generators satisfy the 16 relations given on page 50 of Higman's book:
--------------------------------------------------------------------------
gap> k := ClassTransposition(0,2,1,2);; # kappa in Higman74
gap> l := ClassTransposition(1,2,2,4);; # lambda "
gap> m := ClassTransposition(0,2,1,4);; # mu "
gap> n := ClassTransposition(1,4,2,4);; # nu "
gap> H := Group(k,l,m,n);
<rcwa group over Z with 4 generators>
gap> G = H;
true
gap> HigmanThompsonRels :=
> [ k^2, l^2, m^2, n^2, # (1) in Higman74, p.50.
> l*k*m*k*l*n*k*n*m*k*l*k*m, # (2) "
> k*n*l*k*m*n*k*l*n*m*n*l*n*m, # (3) "
> (l*k*m*k*l*n)^3, (m*k*l*k*m*n)^3, # (4) "
> (l*n*m)^2*k*(m*n*l)^2*k, # (5) "
> (l*n*m*n)^5, # (6) "
> (l*k*n*k*l*n)^3*k*n*k*(m*k*n*k*m*n)^3*k*n*k*n,# (7) "
> ((l*k*m*n)^2*(m*k*l*n)^2)^3, # (8) "
> (l*n*l*k*m*k*m*n*l*n*m*k*m*k)^4, # (9) "
> (m*n*m*k*l*k*l*n*m*n*l*k*l*k)^4, #(10) "
> (l*m*k*l*k*m*l*k*n*k)^2, #(11) "
> (m*l*k*m*k*l*m*k*n*k)^2 ];; #(12) "
gap> Set(HigmanThompsonRels);
[ IdentityMapping( Integers ) ]
--------------------------------------------------------------------------
In fact, G = H is the group which is generated by the set of all class
transpositions which interchange residue classes modulo powers of 2.
Def.: Given a set P of odd primes, let CT_P(Z) be the group which is
generated by all class transpositions (r_1(m_1),r_2(m_2)) for which
all odd prime factors of m_1 and m_2 lie in P.
In this notation, G is the group CT_P(Z), where P = {} (i.e. the empty set).
By Corollary 3.7 in
http://www.cip.mathematik.uni-stuttgart.de/~kohlsn/preprints/simplegp.pdf,
the groups CT_P(Z) are all simple.
The intersection of these uncountably many infinite simple groups is our
group G, hence is isomorphic to the Higman-Thompson group.
All groups CT_P(Z) are subgroups of the group CT(Z), which is generated
by the set of all class transpositions. Thus, very roughly we can depict
the situation as follows:
CT(Z)
/ | \
/ | \
/ | \
/ | \
CT_{3}(Z) ... CT_{5,7,23}(Z) ... CT_{p = 1 mod 4}(Z) ...
\ | /
\ | /
\ | /
\ | /
G (Higman-Thompson group)
Our group G preserves a certain tree structure. The groups CT_P(Z) for
nonempty sets P of odd primes do not do so, which apparently makes
investigating them essentially more difficult -- even if P is finite,
or just {3}, say.
Any ideas, comments, hints, questions, suggestions, ...
are greatly appreciated.
Best wishes,
Stefan Kohl
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