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For the definitions, please see the relevant pages,
accessible through my research page.
Closure operators on a subgroup lattice
 In joint work with Martha Kilpack (SUNY Oneonta), a
characterization of the finite groups $G$ with the property
that the closure operators on the subgroup lattice of $G$ form a
lattice that is isomorphic to a subgroup lattice: this occurs if and
only if $G$ is cyclic of prime power order
 This was extended in 2017 to closure operators on any finite
lattice (the resulting lattice is a subgroup lattice if and only if
the original lattice was a finite chain). In 2018, we completed the
determination in the case of infinite groups as well; the resulting
lattice is isomorphic to a subgroup lattice if and only if $G$ is
either cyclic of prime power order, or isomorphic to the Prufer
$p$group $\mathbb{Z}_{p^{\infty}}$.
Capability of groups
 A generalization of Baer's Theorem: Let k be a positive integer,
and let $C_1,\ldots,C_n$,
be cyclic $p$groups, $p$ a prime with $p\gt k$, listed in
ascending order of size. If $G$ is
the $k$nilpotent product of the $C_i$, then $G$ is capable
if and only if $r\gt 1$ and the orders of the last two cyclic groups
are equal.
 A necessary condition: Let $G$ be a $p$group of class $k\gt 0$,
and let $x_1,\ldots,x_r$ be a minimal generating set. Let the order of $x_i$
be $p^{a_i}$, with $a_1\leq\cdots\leq a_r$. If $G$ is capable, then
$r\gt 1$ and $a_r\lt a_{r1} + \lfloor(k1)/(p1)\rfloor$.
 The condition above is sufficient when $G$ is the
$k$nilpotent product of the cyclic groups generated by the $x_i$,
and $p\leq k$. The inequality is best possible: for every
$k$ and $p$ there is a capable example where we have
equality.
 In the case of capable groups of class 2 and exponent $p$, with $p$
an odd prime, I have proven that given a group $G$, the capability of $G$
is equivalent to an explicit question about linear transformations
between two finite dimensional vector spaces, and thus may be
explicitly checked. Using this equivalence, I have obtained a number
of necessary and of sufficient conditions for capability of $G$.
 A 5generated $p$group of class two and prime exponent is
one and only one if nontrivial cyclic; a direct product amalgamating
a cyclic subgroup of the commutators groups; or capable.
 A characterization of the capable 2generated pgroups of
class two.
Dominions and amalgams in varieties of nilpotent groups
 A complete description of dominions in any variety all of whose
elements are nilpotent groups of class at most 2. This appears in
Amalgams of nilpotent groups of class 2, Journal of Algebra
274 (2004) pp. 163.
 There are nontrivial dominions in the variety of all nilpotent
groups of class at most $k$ for any $k\gt 1$, and in the class
of all nilpotent groups. Examples can be found in Dominions in
varieties of nilpotent groups, Comm. Alg. 28 (2000) no. 3,
pp. 12411270.
 A description of the absolutely closed nilpotent groups relative
to any variety all of whose elements are nilpotent groups of class
two. Again, this occurs in Amalgams of nilpotent groups of class
two, Journal of Algebra 274 (2004) pp. 163.
 A complete characterisation of the weakly and strongly embeddable
amalgams in any variety of nilpotent groups of class two, as well as a
characterisation of the weak, strong, and special amalgamation bases
and related results. Again, in Amalgams of nilpotent groups of
class two.
Dominions in the variety of metabelian groups
 There are nontrivial dominions in the variety of metabelian
groups. In fact, there is a finitely generated group $G$, and a
subgroup $H$
of $G$, which is abelian of finite rank, such that the dominion of $H$
in
$G$ is abelian of infinite rank. The example can be found in
Dominions in varieties of nilpotent groups,
Comm. Alg. 28 (2000) no. 3, pp. 12411270.
 There are nontrivial dominions in the variety of all groups which
are extensions of an abelian group of exponent $n$ by an abelian
group of exponent $m$ (unpublished).
Dominions in varieties generated by simple groups
 By generalizing an argument of B.H. Neumann, we fully
characterize the dominions of subgroups of a finite nonabelian simple
group $S$, in the variety generated by $S$.
This characterization is used to provide many examples of
varieties of groups with nonsurjective epimorphisms. They can be found
in Dominions in varieties generated by simple groups,
Alg. Universalis 48 (2002) pp. 133143.
Behavior of dominions under varietal multiplication
 If $\mathfrak{N}$ is a nontrivial variety which has instances of
nontrivial
dominions, and $\mathfrak{Q}$ is any variety other than the variety of
all groups,
then the varietal product $\mathfrak{NQ}$ also has nontrivial dominions.
 The converse of this is false, as witnessed by the fact,
mentioned above, that the variety of metabelian groups, which is the
product of the variety of abelian groups with itself, has nontrivial
dominions, even though all dominions in the variety of abelian groups
are trivial.
 In consequence, the variety of all solvable groups of length at
most $k$ has nontrivial dominions, as does any product of abelian
varieties with at least two factors.
 We give a lower and upper bound on the dominion of a subgroup $H$
of a group $G$ in the product variety $\mathfrak{NQ}$, in terms of the
laws of $\mathfrak{Q}$, the
dominions in $\mathfrak{N}$, and the internal structure of $G$. Under
mild extra
conditions on $G$, we can describe the dominion exactly.
 All of the above results can be found in Dominions in decomposable
varieties, Alg. Unversalis 43 no. 23 (2000), pp. 217232.
Nonsurjective epimorphisms
 If a varietal product $\mathfrak{NQ}$ has nonsurjective
epimorphisms, then so
does $\mathfrak{N}$.
 In fact, the subgroup $H$ is epimorphically embedded into $G$ in the
variety $\mathfrak{NQ}$ if and only if for every normal subgroup $M$ of
$G$ such that $M\\in \mathfrak{N}$ and $G/M\in \mathfrak{Q}$,
$H\cap M$ is epimorphically
embedded into $M$ (in the variety $\mathfrak{N}$) and $HM=G$.
 If $H$ is epimorphically embedded into a finite nonabelian simple
group $G$ in the variety $\mathfrak{N}$, and $\mathfrak{Q}$ is any
variety other than the variety
of all groups, then $\mathfrak{NQ}$ also has nonsurjective epimorphisms.
 If $H$ is epimorphically embedded into a finite group $G$ in the
variety $\mathfrak{N}$, and $\mathfrak{Q}$ is a finite varietal
product of nilpotent varieties,
each of which contains the infinte cyclic group, then $\mathfrak{NQ}$
has instances
of nonsurjective epimorphisms.
 The above results may be found in Nonsurjective epimorphisms
in decomposable varieties, Alg. Universalis 48 (2002) pp. 145150.
