Date: Mon, 17 Jul 95 15:35:01 MDT
From: moore@santafe.edu (Cris Moore)
To: Werner.Nickel@Math.RWTH-Aachen.DE, drisko@math.berkeley.edu,
hillman@math.washington.edu, rusin@math.niu.edu
Subject: group property
The real reason I'm interested in all this is the following.
I am interested in groups where every element can be uniquely written
x=a_1 a_2 ... a_n
where a_i \in A_i and the A_i are Abelian subgroups (whose intersections
are the identity). Moreover, I'd like the following to be true:
x_1 x_2=(a_1 a_2 ... a_n)(b_1 b_2 ... b_n)=c_1 c_2 ... c_n where c_i \in A_i
and
c_1=a_1 b_1,
c_2=f_2(a_2) g_2(b_2) where f_2 and g_2 are homomorphisms
depending on a_1 and b_1 (i.e. not homomorphisms of a_1 and b_1,
but homomorphisms of a_2 and b_2 which are different depending on
a_1 and b_1)
c_3=f_3(a_3) g_3(b_3) where f_3 and g_2 are homomorphisms depending on
a_1, a_2, b_1, and b_2
...
for instance, for groups with the property I asked about before,
the g_i are all the identity and f_2 is conjugation by a_1,
f_3 is conjugation by a_1 a_2, and so on.
Groups which can be decomposed in this way, and for which the decomposition
has this multiplication rule where the homomorphisms acting on each subgroup
depend only on the previous ones, are related to a particular class of
cellular automata I'm trying to predict.
But so far the only examples I found have just two components, and are
split extensions of their derived subgroup (dihedral groups, A_4, etc.)
So...
1) Which solvable groups have this property?
2) Are there any examples with three or more components?
3) Are there examples where f_i and g_i are other than conjugations and the
identity respectively?
- Cris Moore, Santa Fe Institute moore@santafe.edu
==============================================================================
Date: Mon, 17 Jul 95 16:24:23 MDT
From: moore@santafe.edu (Cris Moore)
To: rusin@math.niu.edu
Subject: Re: group property
What is S_4? The symmetries of the cube?
- Cris
==============================================================================
Date: Mon, 17 Jul 95 19:37:24 MDT
From: moore@santafe.edu (Cris Moore)
To: Werner.Nickel@Math.RWTH-Aachen.DE, drisko@math.berkeley.edu,
hillman@math.washington.edu, rusin@math.niu.edu
Subject: group property
S_4 is another frustrating example ---
the derived series is
S_4 ---> A_4 ---> Z_2^2
| |
Z_2 Z_3
and this is a semidirect product --- there are subgroups
Z_2, Z_3 and Z_2^2 --- but the Z_3 subgroup is not normal with respect to
the Z_2 subgroup, even though Z_3.Z_2^2=A^4 is.
So I still don't know of an example with 3 layers in the derived series,
where the subgroups isomorphic to the quotients are normal with respect
to the quotient subgroups above.
- Cris moore@santafe.edu
==============================================================================
Date: Tue, 18 Jul 95 11:49:08 CDT
From: rusin (Dave Rusin)
To: moore@santafe.edu
Subject: Re: group property
S_4 is another frustrating example ---
the derived series is
S_4 ---> A_4 ---> Z_2^2
| |
Z_2 Z_3
and this is a semidirect product --- there are subgroups
Z_2, Z_3 and Z_2^2 --- but the Z_3 subgroup is not normal with respect to
the Z_2 subgroup, even though Z_3.Z_2^2=A^4 is.
So I still don't know of an example with 3 layers in the derived series,
where the subgroups isomorphic to the quotients are normal with respect
to the quotient subgroups above.
------------------------------------------------------------------------------
Sorry, I didn't know you wanted a chain of subgroups each complemented
in the _whole_ group. Try this.
Let A1 = Z_7 x Z_7. Then the complement will act via some automorphisms
of A1, which form the general linear group GL_2(7). I will in fact
take the complement directly from this automorphism group (that is, I
will decline the opportunity to have a centralizer of A1 besides A1
itself).
Now, within GL_2(7) I can find a split of extension of
abelian groups; for example, there is the "diagonal" subgroup
Aut(Z_7) x Aut(Z_7); since Z_7 is a field, its automorphisms are
the multiplicative group (Z_7)^* = (Z_6). In particular, you can
find automorphisms of order 3, namely multiplication by 2 (mod 7)
or 4 (mod 7). This leads to a subgroup A2 of order 3 in GL(2,7), namely
the one generated by the diagonal matrix diag(2, 4).
Now, the inverse of this matrix is the diagonal matrix diag(4,2), which
has the same eigenvalues, and hence is conjugate to diag(2,4). Indeed,
the two are conjugate via the matrix [ [0,1], [1,0] ]. This matrix
now generates a subgroup A3 of order 2.
So we have abelian groups A1 A2 A3 with A2 and A3 normalizing
A1 and A3 normalizing A2. The semidirect product (A1.A2).A3
of order 2.3.7^2=294 should have the property you want.
More generally, this idea of creating auotmorphism groups in stages
should permit you to construct longer chains of split extensions.
Let me know if you need help with more examples. (I haven't had a
chance to think about the theory yet.)
dave