``Subgroup Lattices of Groups''

by Roland Schmidt

**Ralph Freese**

**8 Feb 1996**

Lattice theory originated late in the last century along two strands. Many of the structures used in logic have lattices, particularly distributive lattices, associated with them. The work of Boole [4] and Schröder [27] are seminal in this regard.

The other strand, which concentrated more on the connections with algebra, was initiated with Richard Dedekind [6] in his studies of ideals in algebraic number fields. The lattices he studied usually satisfied his modular law which is sometimes referred to as Dedekind's law.

Lattice theory was used as a tool for deriving some of the basic structure theorems for group theory and for algebraic systems in general. In [19] and [20] O. Ore gave a purely lattice theoretic proof of the Krull-Schmidt theorem on the uniqueness of direct decompositions, considerably broadening the scope of this result. (See [18] for a clear lattice theoretic proof.) In [21] and [22] he applied lattices, which he called structures, to group theory. R. Baer, J. von Neumann and others employed techniques from lattice theory and projective geometry in proving theorems about groups and other algebraic structures and Marshall Hall's book on group theory contained chapters on both lattice theory and projective geometry.

These results led to the hope that lattice theory might prove to
be a powerful tool in group theory. In the introduction to his
book [28]
Suzuki concluded from his theorem that if *G* is a simple group then
*G* is determined by the lattice of subgroups of ,
that ``we might have a possibility to apply lattice theoretical
considerations to solve the classification problem of finite simple
groups.'' However this hope was not realized;
much more powerful techniques, primarily character theory and
`local analysis,' were used.
Similarly in Abelian group theory
Baer's lattice theory techniques are no longer used. (See page 86
of Kaplansky's monograph [13].)
So group theory and lattice theory went their separate ways. (For
that matter group theory nowadays has little in common with Abelian
group theory.) Group theory had other techniques and lattice theory
had its own deep problems to work on and most of the applications
of lattice theory to algebra were in the field of universal
algebra.

In the last several years some connections between lattice theory
and group theory have resurfaced.
One problem of interest in general algebra: *is every finite
lattice isomorphic to the congruence lattice
of some algebraic
system?* P. Pálfy and P. Pudlák [24]
showed this is closely related to subgroup lattices by reducing the
problem to the question:
*is every finite lattice isomorphic to
an interval sublattice of the lattice of subgroups of a finite
group?*

A good test case is the lattice *M*_{n} which is the lattice
with *n* incomparable elements and a least
and greatest element. *M*_{3} is diagrammed in Figure 1.
If *n*=*q*+1, where *q* is a prime power, then *M*_{n} is the lattice
of subspaces of a vector space *V* of dimension two over the field
with *q* elements and an interval in the lattice of subgroups of
the permutation group on *V* generated by the translations and
scalar multiplication. Can *M*_{n} be an interval in
a subgroup lattice of a finite group for the other values of *n*?

Both group theorists and universal algebraists and
lattice theorists have worked on this
problem. A minimal group whose subgroup lattice has *M*_{n} as an
an interval has a unique minimal normal
subgroup, Köhler [14], and
has no Abelian normal subgroup,
Pálfy and Pudlak [24], assuming *n*-1 is not a prime
power. Feit showed that *M*_{7} and *M*_{11} do occur as intervals in
the subgroup lattice of *A*_{31}; see also Pálfy [23].
The best results are due to
A. Lucchini who has shown that *M*_{n} occurs when
*n*=*q*+2 and when *n*=(*q*^{t}+1)/(*q*+1) + 1, where *q* is a
prime power and *t* is an odd prime [17].
Starting with the O'Nan-Scott Theorem on primitive permutation
groups,
he and R. Baddeley [1, 16]
have reduced the problem of which *M*_{n}'s can be
represented to a list of questions about almost simple groups. In
other words for *M*_{n} to be representable when *n* is not of one of
the above forms implies there is an almost simple group satisfying
strong conditions. Although the problem remains open,
it appears unlikely that these groups exist for all *n*.

Finally we should mention that J. T'u ma has shown that every compactly generated lattice, and hence every finite lattice, is isomorphic to an interval in the subgroup lattice of some (infinite) group [29].

A natural question of particular interest to lattice theorists is:
*are there identities stronger than the modular law true in
all lattices of normal subgroups?*
(By a `law' we mean an equation which is identically true.)
In [9] it is shown that the Arguesian law,
a lattice identity reflecting Desargues law of projective
geometry, holds in all lattices of normal subgroups and in
[8] it is shown that there is no finite basis for the
laws of the class of all lattices of normal subgroups.
P. Pálfy and C. Szabó have exhibited an equation which holds in
all subgroup lattices of Abelian groups but does not hold in all
lattices of normal subgroups. Their proof [25] is
a nice combination of group theory, lattice theory, and
projective geometry.
It is perhaps difficult to see why this result might be unexpected
but recall that a modular lattice which
does not have a sublattice isomorphic to *M*_{3} is
distributive and that if *M*_{3} is embedded into the lattice of normal
subgroups then the quotient group of the top of the *M*_{3} over the
bottom is Abelian. Thus the `modular' parts of the lattice
correspond to the Abelian sections of the group.
See [5] for a survey of results in this
area.

The specific study of the relationship between a group and its lattice of subgroups, the theme of the book under review, began with Ada Rottländer and was championed by R. Baer who spoke on the subject at the 1938 Symposium on Lattice Theory. A nice body of results was proved early on by R. Baer, K. Iwasawa, O. Ore, E. Sadovskiĭ, and M. Suzuki, and Suzuki wrote a monograph on the subject which appeared in 1956. But a great deal of work has been done since then and this is admirably presented in Schmidt's book. One can get an idea of the amount of work in this area since Suzuki's monograph by comparing the lengths of the books: Schmidt's is 572 pages while Suzuki's is under 100.

The main theme of Schmidt's book is the influence group properties have
on the lattices of subgroups and *vice versa*.
For example, which group theoretical properties
are determined by the the lattice of subgroups? Which groups are
determined by their lattice of subgroups and, given a group, which
groups have the same lattice of subgroups.

Let *L*(*G*) denote the lattice of subgroups of a group *G*.
Borrowing the terminology of projective geometry, a lattice
isomorphism from *L*(*G*) onto is called a
*projectivity* from *G* to .

The first chapter introduces the basic notions and some of the
early results proved in this area. Some examples include Ore's theorem
that *L*(*G*) is distributive if and only if *G* is locally cyclic
and its corollary that *G* is cyclic if and only if
*L*(*G*) is distributive and satisfies the ascending chain
condition. Of course this implies that we can identify
elements of the lattice of subgroups corresponding to cyclic
subgroups.
*G* is finite if and only if *L*(*G*) is.
If *L* is a finite lattice which does not have a chain as a direct
factor then there are only finitely many groups having *L* as a
subgroup lattice.

The second chapter studies groups whose subgroup lattice is
modular. These were characterized by K. Iwasawa in the 1940's
for locally finite and nonperiodic groups. Schmidt completes the
characterization by handling the periodic case.
A *Tarski group* is an infinite group all of whose proper subgroups
have prime order. A. Yu. Ol'shanskiĭ
[11, 12]
showed such groups exist and
Schmidt's characterization is in terms of them. Schmidt
also corrects some errors in Iwasawa's proof.
Baer's results about projectivities between Abelian groups are also
presented in this chapter.
Chapter 3 studies groups *G* with *L*(*G*) complemented

Chapters 4, 5, and 6 study projectivities between groups.
Chapters 4 and 5 do this for *G* finite while Chapter 6 covers the
infinite case. Many important properties of groups are defined in
terms of normal subgroups and so it is important to get a handle on
these as much as possible.
An element of a lattice is called *modular* if,
roughly, the modular law holds whenever it is substituted into one
of the variables. Every normal subgroups is a modular element of
the subgroup lattice, and more generally every permutable subgroup
(a subgroup *M* with *MH*=*HM* for every subgroup *H*) is a modular
element.
The image of a normal subgroup under a
projectivity is not necessarily normal, while, of course,
the image of a modular subgroup is modular.

For a subgroup *H* let *H*^{G} be smallest normal subgroup containing
*H* (the *normal closure*)
and *H*_{G} the largest normal subgroup contained in *H* (the
*core*).
Structure theorems for *M*^{G}/*M*_{G} are given in Chapters 5 and 6
for a modular subgroup *M*.
If is a projectivity from *G* to and
, let and be the
normal closure and core of . Then *H* and *K* are normal
subgroups. A detailed analysis shows that *H*/*K* is solvable of
length at most 4 and is solvable of
length at most 5. This analysis is due to G. Busetto,
F. Menegazzo, F. Napolitani, Schmidt, and G. Zacher.
These results are used to give lattice theoretic characterization
of certain classes of finite groups such as simple groups, perfect groups,
solvable groups, and supersolvable groups. More importantly
information about projectivities is obtained.

The situation for infinite groups, studied in Chapter 6, is more
complicated. The notion of modular subgroup must be replaced by
permodular subgroup.
This is a more complicated notion defined in terms of certain
subgroups having finite index and so it is not obvious that it is a
pure lattice property. Fortunately it is possible to characterize
subgroups of finite index lattice theoretically (and thus the
projective image of a subgroup of finite index has finite index).
Using this result and the notion of permodularity many of the results
for finite groups can be extended to infinite groups. For example,
a finite group *G* is simple if and only if and *G* are the
only modular elements of *L*(*G*). The Tarski groups mentioned above
show that this characterization does not extend to infinite simple groups.
Nevertheless a group is simple if and only if and *G* are the
only permodular subgroups.

Lattice theoretical characterizations are also obtained for perfect, hyperabelian, polycyclic, finitely generated solvable (but not for solvable groups in general), hypercyclic, and supersolvable groups.

A group is *determined* by its subgroup lattice if a
projectivity from *L*(*G*) to implies *G* is
isomorphic to ; it is *strongly determined* if, in
addition, any automorphism of *L*(*G*) is induced from an automorphism
of *G*. Chapter 7 gives several classes of groups which are
determined by their subgroup lattices. For example free groups and
non-Abelian torsion-free nilpotent groups are strongly determined.
(These results are due to Sadovskiĭ). Suzuki's result that, if
*G* is a finite simple non-Abelian group, is determined is
generalized to centerless perfect groups.
Using the classification of finite simple groups it
can be shown that a finite simple non-Abelian
group is determined by its lattice of subgroups.

Chapter 8 studies groups for which there is an
isomorphisms of *L*(*G*) onto the dual of some .
Baer began this line of study; he characterized Abelian groups with
this property. Zacher showed that a finite group with this
property must be solvable and its lattice of subgroups is isomorphic
to the lattice of subgroups of an Abelian group. The analysis can
be extended to locally finite groups, but the Tarski groups have
self-dual lattices and the structure of arbitrary groups *G* with
*L*(*G*) isomorphic to the dual of some
is not known.

The last chapter studies the lattice of normal subgroups and other
lattices associated with groups. Of course the lattice of normal
subgroups is much smaller than *L*(*G*) and so tells one much less
about *G*. (From a general algebra viewpoint it would be more
interesting to see what information about *G* can be obtained from
the normal subgroup lattice of *K* for subgroups *K* of
and .)

Schmidt is to be commended on a very nice book with nice collection of interesting results. The presentation is very good. While most of the arguments are group theoretic, they are easily accessible to nonspecialists.

Wed Feb 28 13:55:46 HST 1996