47 research outputs found
Bilinear maps and central extensions of abelian groups
We show that every nilpotent group of class at most two may be embedded in a
central extension of abelian groups with bilinear cocycle. The embedding is
shown to depend only on the base group. Some refinements are obtained by
considering the cohomological situation explicitly.Comment: 16 Pages, Plain Te
Capable groups of prime exponent and class two
A group is called capable if it is a central factor group. We consider the
capability of finite groups of class two and exponent , an odd prime. We
restate the problem of capability as a problem about linear transformations,
which may be checked explicitly for any specific instance of the problem. We
use this restatement to derive some known results, and prove new ones. Among
them, we reduce the general problem to an oft-considered special case, and
prove that a 3-generated group of class 2 and exponent is either cyclic or
capable.Comment: 20 p
Words and Dominions
A necessary and sufficient condition for an element of an algebra (in the
sense of Universal Algebra) to be in the dominion of a subalgebra is given, in
terms of transferable sets. This criterion is then used to formulate a more
wieldy sufficient condition. Finally, some connections to a purely
combinatorial setting are outlined.Comment: Plain TeX, 15 pp. (Universal Algebra
A correction to a result of B. Maier
In a 1985 paper, Berthold J. Maier gave necessary and sufficient conditions
for the weak embeddability of amalgams of two nilpotent groups of class two
over a common subgroup. Then he derived simpler conditions for some special
cases. One of his subsequent results is incorrect, and we provide a
counterexample. Finally, we provide a fix for the result.Comment: Four pages, LaTeX fil
Embedding groups of class two and prime exponent in capable and non-capable groups
We show that if is any -group of class at most two and exponent ,
then there exist groups and of class two and exponent that
contain , neither of which can be expressed as a central product, and with
capable and not capable. We provide upper bounds for in terms of in each case.Comment: 5 pages; title change, minor correction
On the capability of finite groups of class two and prime exponent
We consider the capability of -groups of class two and odd prime exponent.
The question of capability is shown to be equivalent to a statement about
vector spaces and linear transformations, and using the equivalence we give
proofs of some old results and several new ones. In particular, we establish a
number of new necessary and new sufficient conditions for capability, including
a sufficient condition based only on the ranks of and .
Finally, we characterise the capable groups among the 5-generated groups in
this class.Comment: 43 pp; incorporates results from older paper; fix amsrefs/hyperref
incompatibility and a typ
Absolutely closed nil-2 groups
Using the description of dominions in the variety of nilpotent groups of
class at most two, we give a characterization of which groups are absolutely
closed in this variety. We use the general result to derive an easier
characterization for some subclasses; e.g. an abelian group is absolutely
closed in if and only if is cyclic for every prime .Comment: 21 pages plain TeX. Final version, with full classificatio
Amalgamation bases for nil-2 groups of odd exponent
We study the strong, weak, and special amalgamation bases in the varieties of
nilpotent groups of class two and exponent n, where n is odd. The main result
is a characterization of the special amalgamation bases for these varieties. We
also characterize the weak and strong bases. For special amalgamation bases, we
show that there are groups which are special bases in varieties of finite
exponent but not in the variety of all nil-2 groups, whereas for weak and
strong bases we show this is not the case. We also show that in these
varieties, as well as the variety of all nil-2 groups, a group has an absolute
closure (in the sense of Isbell) if and only if it is already absolutely
closed, i.e. if and only if it is a special amalgamation base.Comment: 29 pages, LaTeX with ajour.cls packag
Nonsurjective epimorphisms in decomposable varieties of groups
A full characterization of when a subgroup of a group in a varietal
product is epimorphically embedded in (in the variety ) is given. From this, a result of S.~McKay is derived, which states that
if has instances of nonsurjective epimorphisms, then
also has instances of nonsurjective epimorphisms. Two partial converses to
McKay's result are also given: when~ is a finite nonabelian simple group;
and when~ is finite and is a product of varieties of nilpotent
groups, each of which contains the infinite cyclic group.Comment: Play TeX, 16 p
Dominions in decomposable varieties
Dominions, in the sense of Isbell, are investigated in the context of
decomposable varieties of groups. An upper and lower bound for dominions in
such a variety is given in terms of the two varietal factors, and the internal
structure of the group being analyzed. Finally, the following result is
established: If a variety has instances of nontrivial dominions,
then for any proper subvariety of , also
has instances of nontrivial dominions.Comment: Plain TeX, 22 pp. Typos corrected. Final versio