From: Derek Holt
Date: Thu, 1 Feb 1996 10:17:32 GMT
...stuff deleted...
>
> 2) Does there exist a group G with a unique nontrivial normal subgroup
> H such that both G/H and H are non-abelian ?
> Of course, such a group must have at least 3600 elements;
> does anyone of You know such a group ?
Yes - you get such examples easily by taking wreath products.
The smallest is A5 Wr A5 of order 60^6. It has a unique proper nontrivial
normal subgroup N = A5 x A5 x A5 x A5 x A5, where the factors are
permuted by the conjugation action of G/N = A5.
Derek Holt.
__________________________________________________________________________
From: Bettina Eick
Subject: Re: groups with special lattices of normal subgroups
To: group-pub-forum@maths.bath.ac.uk
Date: Thu, 1 Feb 1996 15:40:04 +1553003 (MET)
Erhard Aichinger asked the following question.
> 1) Is there any characterisation of groups in which the
> lattice of normal subgroups is a chain ?
A finite nilpotent group in which the lattice of normal subgroups
is a chain is a cyclic $p$-group.
However, it seems more complicated to characterise even the fi-
nite soluble groups in which the lattice of normal subgroups is a
chain. It is a necessary, that each complemented chief factor
$M/N$ in such a group $G$ must be a faithful $G/M$-Module. But
this is clearly not sufficient.
By the following method one can construct finite soluble groups
of this type: Let $G$ be a finite soluble group such that its
lattice of normal subgroups is a chain and let $N$ be its unique
minimal normal subgroup. Then $N$ is an elementary abelian
$p$-group and $N$ is generated by a single $G$-conjugacy class of
elements. So by a theorem of Gasch"utz there exists a faithful
represenation of $G$ over $F_q$ for any prime $q <> p$ and the
split extension of $G$ with this faithful $F_q G$-Module has
again the property that the lattice of normal subgroups is a
chain.
Best wishes, Bettina Eick
_________________________________________________________________
Date: Thu, 1 Feb 96 14:50:19 GMT
From: "A.Camina"
Subject: Re: groups with special lattices of normal subgroups
I am sure the right person to ask is Rolandt Schmidt
at Kiel. I believe he has written a book on lattices of
groups.
Alan Camina
_________________________________________________________________
Date: Thu, 1 Feb 96 14:22 +0200
From: MANN@vms.huji.ac.il
Subject: special lattices of normal subgroups
> quotes E.A.
I suggest you look at R.Schmidt's recent book on lattices of subgroups.
>2) Does there exist a group G with a unique nontrivial normal subgroup
> H such that both G/H and H are non-abelian ?
> Of course, such a group must have at least 3600 elements;
Do you want G in 2) to have the property in 1) (the normal subgroups form
a chain)? The smallest I can think of (smaller than Holt's) is the wreath
product of A_5 by S_3, with respect to the natural action of S_3, i.e. there
is a normal subgroup isomorphic to A_5 cross itself three times, and S_3
permutes the three factors. This has order 6.60^3.
Avinoam Mann
_________________________________________________________________
From: Derek Holt
Date: Fri, 2 Feb 1996 16:50:15 GMT
Subject: Re: groups with special lattices of normal subgroups
> 2) Does there exist a group G with a unique nontrivial normal subgroup
> H such that both G/H and H are non-abelian ?
> Of course, such a group must have at least 3600 elements;
> does anyone of You know such a group ?
When I suggested A5 Wr A5 as an example, I imagined that you were asking for
a group with a unique nontrivial (presumably proper) normal subgroup H, where,
in addition, H and G/H are both nonabelian. I still think that this is the
smallest group with this property.
After all, I reason to myself, if you only wanted a group with a unique H
that has that property, then why would you have said "nontrivial", and why
would you have regarded it as clear that such a group would have at least 3600
elements? In fact the two covering groups of S4 have order 48, and their
only normal subgroups have orders 1, 2, 8, 24 and 48 (which form a chain).
Of those, only the one of order 8 has both H and G/H nonabelian.
So what did you really mean?
Derek Holt.
_______________________________________________________________________
Date: Sat, 3 Feb 96 2:20 +0200
From: MANN@vms.huji.ac.il
Subject: apology (re: groups with special lattice...)
Somehow I managed to misread Aichinger's question, and have taken it to mean:
"a group with a unique _minimal_ normal subgroup etc". Of course, he just said
"a unique normal subgroup", and of course Derek Holt is right, his example is
the smallest such. Moreover, I think that, given the structure of the
automorphism groups of the finite simple groups, it follows that the only
finite groups that answer the question are wreath products of two finite simple
non-abelian groups, with respect to some transitive permutation action of the
top factor (I've not checked this carefully).
Well, as long as we are in a pub, let us say that I've taken one too many.
Avinoam Mann
________________________________________________________________________
Date: Mon, 5 Feb 96 13:44:46 +1100
From: Stephen Glasby
Subject: Re: groups with normal subgroup lattice a chain
Erhard Aichinger asked whether there is "... any characterisation
of groups in which the lattice of normal subgroups is a chain ?"
As Bettina Eick stated such a characterisation seems complicated
for finite soluble groups, as suggested by her construction of such
groups in which their chief factors are elementary abelian of coprime
orders.
A further technique for constructing examples occurs in a paper with
R.B. Howlett and myself: Extraspecial towers and Weil representations,
J. Algebra 151 (1992), 236--260.
Here infinite groups are constructed in which the only nontrivial normal
subgroups are terms of the derived series. One example has chief factors
or orders 2,3,2^2,2,3^2,3,2^6,2,3^8,3,... .
At one point I was interested in classifying finite soluble groups
whose composition length (or order) was minimal subject to having
a given derived length. All examples I found had the property that
their normal subgroup lattice was a chain which coincided with its derived
series. While it is clear that such a group must have a unique minimal
normal subgroup, I was unable to prove the normal subgroup lattice is a chain.
It seems unlikely that one can classify the finite soluble groups whose
normal subgroup lattice is a chain, unless one can classify the p-groups
whose lattice of characteristic subgroups is a chain. This latter class
contains many (all?) p-groups of maximal class.
Stephen Glasby
_______________________________________________________________________
Date: Tue, 6 Feb 96 17:27 +0200
From: MANN@vms.huji.ac.il
Subject: special normal subgroup lattice and p-groups
Commenting on Erhard Aichinger's question whether there is "... any
characterisation of groups in which the lattice of normal subgroups
is a chain ?", Stephen Glasby said (among other things):
"It seems unlikely that one can classify the finite soluble groups whose
normal subgroup lattice is a chain, unless one can classify the p-groups
whose lattice of characteristic subgroups is a chain. This latter class
contains many (all?) p-groups of maximal class.
Stephen Glasby"
Not in all p-groups of maximal class the char. subgroups lattice is a
chain, though it is certainly true that many of them, perhaps "most" have
this property. Let G be a p-group of max. class. Then the normal subgroups
of G are its p + 1 maximal subgroups and the terms of the lower central
series. The latter are characteristic and form a chain, so the char. sgp's
of G form a chain iff there is at most one char max sgp. One max sgp is
always char., namely the centraliser of G'/G_4, which is also the centraliser
of all factors G_i/G_{i+2}, for 1 < i < c - 1, where c = cl(G), the nilpotency
class of G. If this centraliser is different from the centraliser of G_{c-1},
then G is called exceptional, and then the char sgps are not a chain.
N.Blackburn showed, that if |G| = p^n (so n = c + 1), then if G is exceptional,
then p > 3, n is even, and 5 < n < p + 2. Conversely, if p and n obey these
restrictions, then exceptional groups of order p^n exist, so they give
examples of groups of max class whose char sgps are not a chain.
An easier example is obtained by looking at 2-groups. The 2-groups of max
class are the dihedral, (generalised) quaternion, and the semi-dihedral groups.
In the first two types the char sgps form a chain, the cyclic max sgp being
the only char max sgp (except for the ordinary quaternion group of order 8,
where no max sgp is characteristic). But in the semi-dihedral group the three
max sgps are non-isomorphic, hence characteristic.
Some information on the automorphism groups og groups of max class is given by
Leedham-Green and McKay, in Quart. J. Math. Oxford (2) 35. Their results imply
that for those groups that they consider the char sgps are a chain, but they
do not consider all groups of max class.
Avinoam Mann
________________________________________________________________________
Date: Wed, 7 Feb 1996 13:20:30 +1100
From: "M.F.(Mike) Newman"
Subject: Characteristic subgroups of p-groups with max class
Briefly, a gloss on what Avi Mann says.
There is a result of Ursula Martin which says that "most"
p-groups have automorphism groups which only act trivially on
the Frattini quotient - this is also true, I believe, for
groups with maximal class - and hence in "most" all the
normal subgroups are characteristic.
Mike (Newman)
______________________________________________________________________
Date: Wed, 7 Feb 96 13:35 +0200
From: MANN@vms.huji.ac.il
Subject: Characteristic subgroups of p-groups
M.F.(Mike) Newman writes:
There is a result of Ursula Martin which says that "most"
p-groups have automorphism groups which only act trivially on
the Frattini quotient - this is also true, I believe, for
groups with maximal class - and hence in "most" all the
normal subgroups are characteristic.
Mike (Newman)
I think that Ursula never published the full proof, only an outline. I've
also heard her talk about that result. As far as I remember, the proof is
by counting and comparing the asymptotic behaviour of the number of all
p-groups to the number of the ones with the given property. Such proof would
say nothing about groups of maximal class, which, for a given order, form
anyway a negligible fraction of the set of all groups of that order
(asymptotically speaking). Of course Mike Newman is right in saying that
Martin's result shows that in most p-groups all maximal subgroups are
characteristic.
By the way, Stephen, do you have other interesting examples of p-groups in
which the char. subgroups form a chain?
Avinoam Mann
----------------------------------------------------------------------
Date: Wed, 7 Feb 96 18:48 MET
From: Eamonn OBrien
Subject: Re: Characteristic subgroups of p-groups
Avinoam Mann asks
> By the way, Stephen, do you have other interesting examples of p-groups in
> which the char. subgroups form a chain?
Here are some other 2-groups which satisfy the chain condition.
1. The group of order 16 obtained by taking a split
extension of C2 x C2 by C4 acting invertingly.
It has the power-commutator presentation:
<1, 2, 3, 4 | 3 = [2, 1], 4 = 1^2, all other relations trivial>
2. The groups D8 Y D8 and D8 Y Q8 of order 32.
3. The subdirect product of two quaternions of order 8 having pcp
<1, 2, 3, 4, 5: 1^2 = 4, 2^2 = 4, 3^2 = 5, [2, 1] = 4, [3, 1] = 5>.
4. The direct product of the dihedral group of order 8
with copies of the cyclic group of order 2.
Similarly for the quaternion group of order 8.
Eight of the 14 groups of order 16 and 17 of the 51 groups
of order 32 have this property. I'm happy to supply
presentations for these on request.
The tables of Hall & Senior provide access to this
information (they list characteristic subgroups) or
it can readily be computed from the 2-groups library.
Eamonn O'Brien
____________________________________________________________
Date: Thu, 8 Feb 96 13:57:01 +1100
From: Stephen Glasby
Subject: Re: Characteristics subgroups forming a chain
Avinoam Mann made a passing comment:
> By the way, Stephen, do you have other interesting examples of p-groups in
> which the char. subgroups form a chain?
The short answer to your question is `No, sorry'. I recall some
isolated examples of maximal class groups, although in some sense
one is most interested in generalizing from specific examples
(such as the ones Eamonn produced) to produce infinite families
of p-groups whose characteristic subgroups form a chain.
Presumably, one can (with effort) construct all such
(finite) abelian p-groups. I have not tried to do so, however,
three examples are the abelian p-groups of type (n,...,n),
(1...,1,2) and (n). (Really the first example subsumes the third.)
As this is a Pub discussion, let me claim, rather boldly, that
the following examples which are `perturbations' of these groups,
also have the CSC property (Characteristics Subgroups form a Chain)
E_{p^{2n+1}} extraspecial gp of order p^{2n+1}
E_{p^{2n+1}} Y C_{p^2} i.e. central product with a a cyclic gp of order p^2
D_{2^n} the dihedral gp of order 2^n
Q_{2^n} the quaternion gp of order 2^n
Please treat all of these claims with scepticism.
Regards, Stephen Glasby
_____________________________________________________________
Date: Thu, 8 Feb 1996 17:04:51 +0100
From: Andreas Caranti
Subject: Re: Characteristic subgroups forming a chain
Stephen Glasby writes:
>Presumably, one can (with effort) construct all such
>(finite) abelian p-groups. I have not tried to do so, however,
>three examples are the abelian p-groups of type (n,...,n),
>(1...,1,2) and (n).
Under the convenient assumption that we're in a pub, and that I've had one
pint (we're not going metric here, huh?) too many...
Let $G$ be a finite Abelian $p$-group with CSC (Characteristic Subgroups
form a Chain). Suppose the minimum order of a cyclic direct factor of $G$
is $p^{m}$. Consider the two characteristic subgroups $G^{p}$ and
$\Omega_{m}(G)$ of $G$. Clearly $\Omega_{m}(G) \not\le G^{p}$. Thus $G^{p}
\le \Omega_{m}(G)$, so that $G^{p^{m+1}} = 1$. There are two possibilities:
1. $G$ has exponent $p^{m}$. Here $G$ is homocyclic, a case already noted
by Stephen.
2. $G$ has exponent $p^{m+1}$, so the cyclic direct factors of $G$ are of
order $p^{m}$ or $p^{m+1}$ here (both occur). I haven't checked all details
here, but I'd say ("Please treat all of these claims with scepticism."
S.G.) that such a group has CSC too, and generalizes the case $(1, \dots,
1, 2)$ mentioned by Stephen.
What do you think?
Andreas Caranti
____________________________________________________________________
Date: Fri, 9 Feb 1996 13:37:01 +0100
From: Andreas Caranti
Subject: Characteristic maximal subgroups in groups of maximal class
Avinoam Mann has noted that in the "exceptional" $p$-groups of maximal
class there are two distinct characteristic maximal subgroups, so such
groups do not have CSC (Characteristic Subgroups form a Chain).
Now it is well-known (see B. Huppert, Endliche Gruppen I, III.14.13 and
III.14.23) that in a $p$-group $G$ of maximal class of class $c$, all
maximal subgroups distinct from $C_{G} (G_{2} / G_{4})$ and $C_{G}
(G_{c-1})$ are of maximal class themselves.
So let $M = \left< x, G_{2} \right>$ be one of these maximal subgroups. If
$y \in G \setminus M$, clearly $[y, x] = z \in G_{2} \setminus G_{3}$. We
also have $G_{3} = M_{2}$. Therefore $y$ induces on $M$ an automorphism
which permutes the $p$ maximal subgroups of $M$ distinct from $G_{2}$. So
$G_{2}$ is the only characteristic maximal subgroup of $M$, and $M$ has
CSC.
In other words, maximal subgroups of groups of maximal class that are of
maximal class themselves have CSC.
Andreas Caranti