Algebra - Algebra - Algebra - Algebra - Algebra - Algebra - Algebra - Algebra - Algebra - Algebra - Algebra - Algebra - Algebra - Algebra - Algebra - Algebra - Algebra

(with R. Göbel) , submitted

We consider the monoid Inj(

**
Automorphism groups of totally ordered sets: a retrospective survey
**

(with V. V. Bludov,
A. M. W. Glass)
, Mathematica Slovaca 61 (2011), 373-388.

__Abstract.__

In 1963, W. Charles Holland proved that every lattice-ordered
group can be embedded in the lattice-ordered group of all order-
preserving permutations of a totally ordered set. In this article
we examine the context and proof of this result and survey some
of the many consequences of the ideas involved in this important
theorem.

**
On extension of coverings
**

(with I. Rivin)
, Bull. London Math. Soc. 42 (2010), 1044-1054.

__Abstract.__

We address the question of when a covering of the boundary of a
surface F can be extended to a covering of the surface (equivalently: when is th
ere
a branched cover with a prescribed monodromy ). If such an extension is possible
,
when can the total space be taken to be connected? When can the extension be
taken to be regular? We give necessary and sucient conditions for both finite an
d
infinite covers (infinite covers are our main focus). In order to prove our resu
lts,
we show group-theoretic results of independent interests, such as the following
extension (and simplification) of the theorem of Ore [11]: every element of the
infinite symmetric group is the commutator of two elements which, together, act
transitively.

**
On full groups of measure preserving and ergodic transformations with uncountable cofinalities
**

(with C.Holland, G.Ulbrich) Bulletin London Math. Soc. 40 (2008), 463-472.

__Abstract.__

The group of all measure preserving permutations of the unit interval
and the full group of an ergodic transformation of the unit interval
are shown to have uncountable cofinality and the Bergman property.
Here, a group *G* is said to have the Bergman property, if for
any generating subset *E* of *G*, already some bounded power
of *E* ∪ *E ^{-1}* ∪
{1} covers

**
Uncountable cofinalities of automorphism groups of linear and partial orders
**

(with J. K. Truss)
, Algebra Universalis 62 (2009), 75-90.

__Abstract.__

We demonstrate the uncountable cofinality of the automorphism groups
of various linear and partial orders. We also relate this to the 'Bergman'
property, and discuss cases where this may fail even though the cofinality
is uncountable.

**
Stabilizers of direct composition series**

(with R.Göbel) Algebra Universalis 62 (2009), 209-237

__Abstract.__

Let *R* be a domain, *V* a left *R*-module, and *L*
a composition series of direct summands
of *V* . Our main results show that if *U* is a stabilizer
group of *L* containing the McLaingroup
associated with *L*, then *U* determines the chain (L,⊆)
uniquely up to isomorphism
or anti-isomorphism.

**
Construction of some uncountable 2-arc-transitive bipartite graphs
**

(with R.Gray, J.K.Truss) Order 25 (2008), 349-357.

__Abstract.__

We give various constructions of uncountable arc-transitive bipartite
graphs employing techniques from partial orders, starting with the
cyclefree case, but generalizing to cases where this may be violated.

**
Absolute graphs with prescribed endomorphism monoids
**

(with R.Göbel, S.Pokutta) Semigroup Forum 76 (2008), 256-267, Full version: pdf

__Abstract.__

We consider endomorphism monoids of graphs. It is well-known that any
monoid can be represented as the endomorphism monoid *M* of some
graph Γ with countably many colors. We give a new proof of this
theorem such that the isomorphism between the endomorphism monoid
End(Γ) and *M* is absolute, i.e. End(Γ) ≅ *M*
holds in any generic extension of the given universe of set theory. This
is true if and only if |*M*| , |Γ| are smaller than the first
Erdös cardinal (which is known to be strongly inaccessible). We will
encode Shelah's absolutely rigid family of trees [15] into Γ. The
main result will be used to construct fields with prescribed absolute
endomorphism monoids, see [8].

**
Bifinite Chu Spaces
**

(with G.-Q. Zhang)
in: 2nd Conf. on Algebra and Coalgebra in Computer Science (CALCO), Lecture Notes in Comp. Science vol. 4624, Springer, 2007, pp. 179-193.

__Abstract.__

This paper studies colimits of sequences of finite Chu spaces and their
ramifications. We consider three bases categories of Chu spaces: the generic Chu spaces
(**C**), the extensional Ch spaces (**E**), and the biextensional Chu spaces (**B**). The main
results are: (1) a characterization of monics in each of the three categories; (2) existence
(or the lack thereof) of colimits and a characterization of finite objects in each of the
corresponding categories using monomorphisms/injections (denoted as **iC**, **iE**, and **iB**,
respectively); (3) a formulation of bifinite Chu spaces with respect of **iC**; (4) the existence of universal, homogeneous Chu spaces in this category. Unanticipated results driving
this development include the fact that: (a) in **C**, a morphism (*f*,*g*) is monic iff *f* is injective (but *g*
is not necessarily surjective); (b) while colimits always exist in **iE**, it is not the case for
**iC** and **iB**; (c) not all finite Chu spaces (considered set-theoretically) are finite objects
in their categories. This study opens up opportunities for further investigations into recursively defined Chu spaces, as well as constructive models of linear logic.

**
Normal Subgroups of B _{u}Aut(Ω)
**

(with W.Ch. Holland), Applied Categorical Structures 15 (2007), 153-162

Aut(Ω) denotes the group of all order preserving permutations of the totally ordered set Ω, and if

(with R. Göbel),

We will discover a sufficient criterion for certain permutation groups $G$ to have uncountable strong cofinality, i.e. they can not be expressed as the union of a countable, ascending chain $(H_i)_{i\in\o}$ of proper subsets $H_i$ such that $H_iH_i\subseteq H_{i+1}$ and $H_i=H_i^{-1}$. This is a strong form of uncountable cofinality for $G$, where each $H_i$ is a subgroup of $G$. This basic tool comes from a nice, recent article by G. M. Bergman on generating systems of the infinite symmetric groups, which we discuss in the introduction. Our main result is a theorem which can be applied to various classical groups including the symmetric groups and homeomorphism groups of Cantor's discontinuum, the rationals, and the irrationals, respectively. They all have uncountable strong cofinality. So the result also unifies various known results about cofinalities. Our favored example is the group \bsym of all bounded permutations of the rationals $\Q$ which has uncountable cofinality but countable strong cofinality. It is discussed in detail.

(with W. Ch. Holland), Forum Mathematicum 17 (2005), 699-710.

Let (/Omega, /leq) be any doubly homogeneous chain and Aut(/Omega) its group of order-automorphisms. We show that if J is a set of generators of Aut(/Omega), then there is a positive integer n such that every element of Aut(/Omega) is a product of at most n members of J/cup J^{-1}. Also, Aut(/Omega) cannot be written as the union of a countable chain of proper subgroups of Aut(/Omega).

(with D. Kuske),

Erdös & Rényi gave a probabilistic construction of the countable universal homogeneous graph. We extend their result to more general structures of first order predicate calculus. Our main result shows that if a class of countable relational structures

**Outer automorphism groups
of ordered permutation groups.**

(with S. Shelah), *Forum
Mathematicum *14 (2002), 605 - 621.

__Abstract.__

An infinite linearly ordered set (*S*,<)is called doubly homogeneous if its automorphism group
*A*(*S*) acts 2-transitively on it. We show that any group *G
*arises as outer automorphism group $*G \cong *Out(*A*(*S*))$
of the automorphism group *A*(*S*), for some doubly
homogeneous chain (*S*,<).

**All groups are outer
automorphism groups of simple groups.**

(with
M. Giraudet and R. Göbel), *Journal London Math. Soc. *64
(2001), 565 - 575.

__Abstract.__

We will show that each group is the outer
automorphism group of a simple group. Surprisingly, the proof is mainly
based on the
theory of ordered or relational structures and their symmetry groups.
By
a recent result of Droste and Shelah (2000) any group is the outer
automorphism
group Out(Aut *T*) of the automorphism group Aut *T* of a
doubly
homogeneous chain (*T*,<). But Aut *T* is never simple.
However,
following recent investigations on automorphism groups of circles, we
are
able to turn (*T*,<) into a circle *C* such that

$Out(Aut *T*) \cong Out(Aut *C*)$. The
unavoidable normal subgroups in Aut *T* evaporate in Aut *C*,
which is now simple, and our result follows.

**On the homeomorphism
groups of Cantor's discontinuum and the spaces of rational and of
irrational numbers.**

(with R. Göbel),
*Bull. London Math. Soc*. 34 (2002), 474 - 478.

__Abstract.__

We will show that the homeomorphism groups of
Cantor's discontinuum, the rationals, and the irrationals, respectively
have uncountable cofinality. It is well known that the homeomorphism
group of Cantor's

discontinuum is isomorphic to the automorphism
group Aut*B* of the countable, atomless boolean algebra *B*.
So also Aut*B* has uncountable cofinality, which answers a
question of Droste and Macpherson. The cofinality of a group *G*
is the cardinality of the
length of a shortest chain of proper subgroups terminating at *G*.

**Rigid chains admitting
many embeddings.**

(with J.K. Truss), *Proc.
Amer. Math. Soc.*, 129 (2001), 1601 - 1608.

__Abstract.__

A chain (linearly ordered set) is *rigid *if
it has no non-trivial automorphisms. The construction of dense rigid
chains was carried out by Dushnik and Miller for subsets of **R**,
and there
is a rather different construction of dense rigid chains of cardinality
$\kappa$*, *an uncountable regular cardinal, using stationary
sets as 'codes', which was adapted by Droste to show the existence of
rigid measurable spaces. Here we examine the possibility that
nevertheless there could be many order-*embeddings* of the chain,
in the sense that the whole chain can be embedded into any interval. In
the case of subsets of **R**, an argument involving Baire category
is used to modify the original one. For uncountable regular cardinals,
a more complicated version of the corresponding argument is used, in
which the stationary sets are replaced by sequences of stationary sets,
and the chain is built up using a tree. The construction is also
adapted to the case of singular cardinals.

**Complementary closed
relational clones are not always Krasner clones.**

(with D. Kuske, R. McKenzie
and R. Pöschel), *Algebra
Universalis* 45 (2001), 155 - 160.

__Abstract.__

In ZFC, it is shown that every relational clone on
a set *A* closed under complementation is a Krasner clone if and
only if *A* is at most countable. This is achieved by solving an
equivalent problem on locally invertible monoids: A partially ordered
set is constructed whose endomorphism monoid is not contained in the
local closure of its
automorphism group.

**The automorphism group of
the universal distributive lattice.**

(with D. Macpherson), *Algebra
Universalis* 43 (2000), 295 - 306.

__Abstract.__

We describe the normal subgroup lattice of the
automorphism groups of the countable universal homogeneous distributive
lattice and of the countable atomless generalized Boolean lattice.
Also, we show that subgroups of these automorphism groups of index less
than continuum lie between the pointwise and the setwise stabilizer of
a finite set.

**On homogeneous
semilattices and their automorphism groups.**

(with D. Kuske and
J.K. Truss), *Order* 16 (1999), 31 - 56.

__Abstract.__

We show that there are just countably many
countable homogeneous semilattices and give an explicit description of
them. For
the countable universal homogeneous semilattice we show that ist
automorphism group has a largest proper nontrivial normal subgroup.

**Simple automorphism groups
of cycle-free partial orders.**

(with J.K. Truss and
R. Warren), *Forum Mathematicum 11* (1997), 279 - 294.

__Abstract.__

The purpose of this paper is to show that the
automorphism groups of many of the 'cycle-free' partial orders studied
are simple. This contrasts strongly with the situation for trees, of
which they form a natural generalization. It was shown that the
automorphism group of any sufficiently transitive tree has at least
$2^{2^{\aleph_0}}$ normal subgroups. All the infinite cycle-free
partial orders studied have simple automorphism groups. The finite
chain case is more involved; where the ordering on chains of the
Dedekind-MacNeille completion can be expressed as a lexicographic
product by a non-trivial discrete (transitive) ordering (respected by
the group), the automorphism group is not simple. For both finite and
infinite chain cases the simple automorphism groups split into two
classes: those where there
is a bound (<12) on the number of conjugates required to express one
non-identity
element in terms of another, and those in which there is no such bound.

**Set-homogeneous graphs and
embeddings of total orders.**

(with M. Giraudet
and D. Macpherson), *Order* 14 (1997), 9 - 20.

__Abstract.__

We construct uncountable graphs in which any two
isomorphic subgraphs of size at most 3 can be carried one to the other
by
an automorphism of the graph, but in which some isomorphism between
2-element
subsets does not extend to an automorphism. The corresponding
phenomenon
does not occur in the countable case. The construction uses a suitable
construction
of infinite homogeneous coloured chains.

**Set-homogeneous graphs.**

(with M. Giraudet,
D. Macpherson and N. Sauer), *J. Combinatorial Theory Ser. B*, 62
(1994), 63 - 95.

__Abstract.__

We investigate set-homogeneity (a weakening of
Fraissé's notion of homogeneity), give an example of a
set-homogeneous graph which is not homogeneous, and characterize it by
its symmetry properties. A variant of Fraissé's amalgamation
theorem is also given.

**Representations of free
lattice-ordered groups.**

*Order* 10
(1993), 375 - 381.

__Abstract.__

We show for any uncountable cardinal $\eta$ that
the free group $*G*_\eta$ of rank $\eta$ has a linear right
ordering
on which the natural action of free lattice-ordered group $*F*_\eta$
of rank $\eta$ is faithful and pathologically 2-transitive. As a
consequence, we obtain results on the root system of prime subgroups of
$*F*_\eta$. This generalizes previous results of McCleary which
required the generalized continuum hypothesis and $\eta$ to be regular.

**On k-homogeneous posets
and graphs.**

(with D.
Macpherson), *J. Combinatorial Theory* Ser. A56 (1991), 1 - 15.

__Abstract.__

A relational structure *A* is called *k*-homogeneous
if each isomorphism between two *k*-element substructures of *A
*extends to an automorphism of *A*. We show that if a
countable
poset $(\Omega,\leq)$ is *1*- or *4*-homogeneous, then it
is
*k*-homogeneous for each $*k*\in **N**$. There are
infinitely
many examples of $\aleph_0$-categorical universal countable posets
$(\Omega,\leq)$
showing that here the number *4* may not be replaced by *2*
or
*3*. We also show that for every $*k*\in **N**$ there are
continuously
many countable $\aleph_0$-categorical universal graphs which are *k*-homogeneous
but not (*k+1*)-homogeneous; this answers a question of R.
Fraissé.

**The root system of prime
subgroups of a free lattice-ordered group (without G.C.H.).**

(with S.H.
McCleary), *Order* 6 (1989), 305 - 309.

__Abstract.__

For the free lattice-ordered group $*F*_\eta$
of rank $\eta$ the root system $*P*_\eta$ of prime subgroups has
been described in considerable detail by McCleary, for
$\eta\leq\omega_0$ and
(with G.C.H.) for regular $\eta>\omega_0$. Here, almost all of that
description is obtained without G.C.H., and most of it without
regularity.

**Embeddings into simple
lattice-ordered groups with different first order theories.**

(with M. Giraudet), *Forum
Mathematicum* 1 (1989), 315 - 321.

__Abstract.__

We show that any lattice-ordered group *G *can
be embedded into continuously many simple divisible lattice-ordered
groups $*H*_*i*$ which are pairwise elementarily inequivalent
both as
groups and as lattices. Moreover, the *l*-groups $*H*_*i*$
can be chosen to have a representation as doubly transitive groups of
order-preserving permutations of chains and such that their group
structure and their lattice-plus-identity structure are mutually
definable in each other withoutadditional parameters.

**Automorphism groups of
infinite semilinear orders (II).**

(with W.C. Holland
and D. Macpherson), *Proc. London Math. Soc.* 58 (1989), 479 -
494.

__Abstract.__

This paper and its predecessor examine certain
infinite semilinear orders ('trees') and their automorphism groups.
Here we classify weakly 2-transitive trees up to
$L_\infty_\omega$-equivalence, and countable weakly 2-transitive trees
up to isomorphism. Various results are obtained about the automorphism
groups, concerning torsion, divisibility, and subgroups of small index.
The automorphism group of some related treelike struktures and their
normal subgroup lattices are also examined.

**Automorphism groups of
infinite semilinear orders (I).**

(with W.C. Holland
and D. Macpherson), *Proc. London Math. Soc.* 58 (1989), 454 -
478.

__Abstract.__

Results are obtained concerning normal subgroups
of the automorphism groups of certain infinite trees. These structures
are mostly $\aleph_0$-categorical, and are trees in a poset-theoretic
but not graph-theoretic sense. It is shown that the automorphism group
has a smallest non-trivial normal subgroup, a largest proper normal
subgroup, and at least $2^2^\omega$ normal subgroups between these two.
We also obtain and use some
results on groups of automorphisms of chains.

**The existence of rigid
measurable spaces.**

*Topology and its
Applications* 31 (1989), 187 - 195.

__Abstract.__

For each uncountable cardinal *k* we
construct $2^*k*$ dense unbounded chains $(*S_i,*\leq)$ of
cardinality *k* which as topological spaces (endowed with the
order-topology) have, in particular, the following properties: They are
each 0-dimensional and mono-rigid, i.e. the only embedding of $*S*_*i*$
into itself is the identity, and they are pairwise nonembeddable into
each other. If $*k*\geq\aleph_2$, the sets $*S*_*i*$ can
be chosen such that, in addition, each
$*G*_\delta$-set is open and hence the measurable spaces $(*S_i,B_i*)$,
where $*B_i*$ is the $\delta$-algebra of all clopen subsets of $*S_i*$,
are mono-rigid and pairwise nonembeddable into each other.

**Super-rigid families of
strongly Blackwell spaces.**

*Proc. Amer. Math.
Soc.* 103(1988), 803 - 808.

__Abstract.__

We construct a complete subfield *F* of *P*(**R**),
isomorphic to *P*(**R**), of pairwise non-Borel-isomorphic
rigid strong Blackwell subsets of **R **such that there are only
'very few'
measurable functions between any two members of *F*. As a
consequence,
we obtain large chains and antichains of non-isomorphic rigid strong
Blackwell
subsets of **R**. Also, there is a collection of continuously many
dense
subsets of **R** such that any two of them differ only by two
elements,
but none of them is a continuous image of any other.

**Normal subgroups and
elementary theories of lattice-ordered groups.**

*Order* 5
(1988), 261 - 273.

__Abstract.__

We show that any lattice-ordered group (*l*-group)
*G *can be *l*-embedded into continuously many *l*-groups
$*H_i*$ which are pairwise elementarily inequivalent both as
groups and as lattices with constant *e*. Our groups $*H_i*$
can be distinguished by group-theoretical first-order properties which
are induced by lattice-theoretically 'nice' properties of their normal
subgroup lattices. Moreover. they can
be taken to be 2-transitive automorphism groups $*A*(*S_i*)$
of
infinite linearly ordered sets $(*S_i*,\leq)$ such that each group
$*A*(*S_i)*$ has only inner automorphisms. We also show that
any countable *l*-group *G *can be *l*-embedded into
a countable *l*-group *H* whose normal subgroup lattice is
isomorphic to the lattice of all ideals of the countable dense Boolean
algebra *B*.

**Squares of conjugacy
classes in the infinite symmetric groups.**

*Trans. Amer.
Math. Soc.* 303 (1987), 503 - 515.

__Abstract.__

Using combinatorial methods, we will examine
squares of conjugacy classes in the symmetric groups $*S*_\nu$ of
all permutations of an infinite set of cardinality $\aleph_\nu$. For
arbitrary permutations $*p*\in *S*_\nu$, we will
characterize when each element $*s*\in *S*_\nu$ with finite
support can be written as a product of two conjugates of *p, *and
if *p *has infinitely many fixed points, we determine when all
elements of $*S*_\nu$ are products of two conjugates of *p*.
Classical group-theoretical theorems are obtained from similar results.

**Completeness properties of
certain normal subgroup lattices.**

*European J.
Combinatorics* 8 (1987), 129 - 137.

__Abstract.__

We examine the normal subgroup lattice of
2-transitive automorphism groups $*A*(\Omega)$ of infinite
linearly ordered sets $(\Omega,\leq)$. Using combinatorial methods, we
prove that in each of these lattices the partially ordered subset of
all those elements which are finitely generated as normal subgroups is
a lattice in which infima and suprema of subsets of cardinality
$\leq\aleph$, always exist; two infinite distributive identities are
also shown to hold. Similar methods are used to give a completeness
result for reduced products of partially ordered sets.

**On the universality of
systems of words in permutation groups.**

(with S. Shelah), *Pacific
J. Math.* 127 (1987), 321 - 328.

__Abstract.__

In the classes of infinite symmetric groups, their
normal subgroups, and their factor groups, we determine those groups
which are equivalent in the sense that they may not be distinguished by
the solvability of a system of finitely many equations in variables and
parameters.

**Partially ordered sets
with transitive automorphism groups.**

*Proc. London
Math. Soc.* 54 (1987), 514 - 543.

__Abstract.__

In this paper, we study the structure of partially
ordered sets $(\Omega,\leq)$ under suitable transitivity assumptions on
their group $*A*(\Omega)$ of all order-automorphisms of
$(\Omega,\leq)$.
Let us call $*A*(\Omega)$ *k*-transitive (*k*-homogeneous)
if whenever *A,B *are two isomorphic subsets of $\Omega$ each
with *k*
elements, then some (any) isomorphism from $(*A*,\leq)$ onto $(*B,*\leq)$
extends to an automorphism of $\Omega$, respectively. We show that if $*k*\geq
4 (*k*=3), there are precisely *k* (5) non-isomorphic
countable partially ordered sets $(\Omega,\leq)$ not containing the
pentagon such
that $(*A*(\Omega)$ is *k*-transitive but not *k*-homogeneous;
if *k*=2, there are a unique countable, and many different
uncountable
sets $(\Omega,\leq)$ of this type. We also give necessary and
sufficient
conditions for two partially ordered sets $(\Omega,\leq)$ not
containing
the pentagon and with *k*-transitive automorphism group $(*k*\geq
2)$ to be $L_\infty_\omega$-equivalent.

**Complete embeddings of
linear orderings and embeddings of lattice-ordered groups.**

*Israel J. Math.*
56 (1986), 315 - 334.

__Abstract.__

An infinite linearly ordered set $(*S*,\leq)$
is called doubly homogeneous, if its automorphism group
Aut$(\Omega,\leq)$ acts 2-transitively on it. We study embeddings of
linearly ordered sets
into Dedekind-completions of doubly homogeneous chains which preserve
all
suprema and infima, and obtain necessary and sufficient conditions for
the
existence of such embeddings. As one of several consequences, for each
lattice-ordered group *G* and each regular uncountable cardinal $** _{k}**\geq|

**On the universality of
words for the alternating groups.**

*Proc. Amer. Math.
Soc.* 96 (1986), 18 - 22.

__Abstract.__

We prove the following theorem on the finite
alternating groups $*A_n*$: For each pair (*p,q*) of nonzero
integers there exists an integer *N*(*p,q*) such that, for
each $*n*\geq *N*$, any even permutation $*a*\in *A_n*$
can be written
in the form $*a=b^p*\cdot *c^q*$ for some suitable elements $*b,c*\in
*A_n*$. A similar result is shown to be true for the finite
symmetric groups $*S_n*$ provided that *p* or *q* is
odd.

**The normal subgroup
lattice of 2-transitive automorphism groups of linearly ordered sets.**

*Order* 2
(1985), 291 - 319.

__Abstract.__

Using combinatorial and model-theoretic means, we
examine the structure of normal subgroup lattices $*N*(*A*(\Omega))$
of 2-transitive automorphism groups $*A*(\Omega)$ of infinite
linearly ordered sets $(\Omega,\leq)$. Certain natural sublattices of $*N*(*A*(\Omega))$
are shown to be Stone algebras, and several first order properties of
their dense and dually dense elements are characterized within the
Dedekind-completion $(\bar\Omega,\leq)$ of $(\Omega,\leq)$. As a
consequence, $*A*(\Omega)$ has either precisely 5 or at least
$2^2^{\aleph_1}$ (even maximal) normal subgroups, and various other
group- and lattice-theoretic results follow.

**A construction of all
normal subgroup lattices of 2-transitive automorphism groups of
linearly ordered sets.**

(with S. Shelah), *Israel
J. Math.* 51 (1985), 223 - 261.

__Abstract.__

We give a complete classification and construction
of all normal subgroup lattices of 2-transitive automorphism groups
$A(\Omega)$ of linearly ordered sets $(\Omega,\leq)$. We also show that
in each of
these normal subgroup lattices the partially ordered subset of all
those
elements which are finitely generated as normal subgroups forms a
lattice
which is closed under even countably-infinite intersections, and we
derive
several further group-theoretical consequences from our classification.

**Normal subgroups of doubly
transitive automorphism groups of chains.**

(with R.N. Ball), *Trans.
Amer. Math. Soc.* 290 (1985), 647 - 664.

__Abstract.__

We characterize the structure of the normal
subgroup lattice of 2-transitive automorphism groups $*A*(\Omega)$
of
infinite chains $(\Omega,\leq)$ by the structure of the Dedekind
completion $(\bar\Omega,\leq)$ of the chain $(\Omega,\leq)$. As a
consequence we obtain various group-theoretical results on the normal
subgroups of $*A*(\Omega)$, including that any proper subnormal
subgroup of $*A*(\Omega)$ is indeed normal and contained in a
maximal proper normal subgroup of $*A*(\Omega)$, and that $*A*(\Omega)$
has precisely 5 normal subgroups if and only if the coterminality of
the chain $(\Omega,\leq)$ is countable.

**Cubes of conjugacy classes
covering the infinite symmetric group.**

*Trans. Amer.
Math. Soc.* 288 (1985), 381 - 393.

__Abstract.__

Using combinatorial methods, we prove the
following theorem on the group *S* of all permutations of a
countably-infinite set: Whenever $*p*\in *S*$ has infinite
support without being
a fixed-point-free involution, then any $*s*\in *S*$ is a
product of three conjugates of *p*. Furthermore, we present
uncountably many new conjugacy classes *C* of *S*
satisfying that any $*s*\in *S*$ is a product of two
elements of *C*. Similar results are
shown for permutations of uncountable sets.

**Products of conjugacy
classes of the infinite symmetric group.**

*Discrete Math.*
47 (1983), 35 - 48.

__Abstract.__

Using combinatorial methods, we will examine
products of conjugacy classes in the symmetric group $*S_0*$ of
all permutations of a countably infinite set. If $*p*\in *S_0*$
has at least one infinite orbit in the underlying set and if $*s*\in
*S_0*$, we give a characterization of when *s* is a product
of two conjugates
of *p*. From this, we derive that if four permutations $*p_i*\in
*S_0* (*i*=1,2,3,4)$ are given which all have infinite
support,
then any permutation of $*S_0*$ is a product of four elements
conjugate
to $*p_1*, *p_2*, *p_3*$ and $*p_4*$,
respectively.
Similar results for permutations of uncountable sets are shown and
classical
group-theoretical results are obtained from these theorems.

**Products of conjugate
permutations.**

(with R.
Göbel), *Pacific J. Math.* 94 (1981), 47 - 60.

__Abstract.__

Using combinatorial methods, we will prove the
following theorem on the permutation group $*S_0*$ of a countable
set: If a permutation $*p*\in *S_0*$ contains at least one
infinite cycle then any permutation of $*S_0*$ is a product of
three permutations each conjugate to *p*. Similar results for
permutations of uncountable sets are shown and classical
group-theoretical results are derived from this.

**Structure of partially
ordered sets with transitive automorphism groups.**

*Memoirs Amer.
Math. Soc.* 334 (1985), reprinted and updated 2003, in
preparation.

__Abstract.__

In this paper, we study the structure of infinite
partially ordered sets $(\Omega,\leq)$ under suitable transitivity
assumptions on their group $*A*(\Omega)=Aut(\Omega\leq)$ of all
order-automorphisms of $(\Omega,\leq)$.

Let $k\in **N**$. We call $*A*(\Omega)$ *k*-transitive
(*k*-homogeneous) if whenever $*A,B*\subseteq \Omega$ are two
subsets of $\Omega$ each with *k* elements and $\varphi: (*A*,\leq)
\to (*B*,\leq)$ is an isomorphism, then there exists an
automorphism $\alpha\in *A*(\Omega)$ which maps *A* onto *B*
(which extends $\varphi$), respectively. $*A*(\Omega)$ is
$\omega$-transitive ($\omega$-homogeneous), if $*A*(\Omega)$ is *k*-transitive
(*k*-homogeneous) for each $*k*\in **N**$.

We show that under the assumption that $*A*(\Omega)$
is *k*-transitive or *k*-homogeneous for some $2\leq *k*\in
**N**$ various sufficiently complicated structures $(\Omega,\leq)$
exist, and we give a classification and characterization of these
structures.
As one of many consequences we obtain that for each $*k*\geq 2$, *k*-transitivity
of $*A*(\Omega)$ is indeed weaker than *k*-homogeneity, but,
surprisingly, for any partially ordered set $(\Omega,\leq)$, $*A*(\Omega)$
is $\omega$-transitive iff $A(\Omega)$ is $\omega$-homogeneous.

last modified December 19, 2012,