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 $p$, $p$ 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 $p$ 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

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

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 $G$ is any $p$-group of class at most two and exponent $p$, then there exist groups $G_1$ and $G_2$ of class two and exponent $p$ that contain $G$, neither of which can be expressed as a central product, and with $G_1$ capable and $G_2$ not capable. We provide upper bounds for ${\rm rank}(G_i^{\rm ab})$ in terms of ${\rm rank}(G^{\rm ab})$ 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 $p$-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 $G/Z(G)$ and $[G,G]$. 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 $G$ is absolutely closed in ${\cal N}_2$ if and only if $G/pG$ is cyclic for every prime $p$.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 $H$ of a group $G$ in a varietal product ${\cal NQ}$ is epimorphically embedded in $G$ (in the variety ${\cal NQ}$) is given. From this, a result of S.~McKay is derived, which states that if ${\cal NQ}$ has instances of nonsurjective epimorphisms, then ${\cal N}$ also has instances of nonsurjective epimorphisms. Two partial converses to McKay's result are also given: when~$G$ is a finite nonabelian simple group; and when~$G$ is finite and ${\cal Q}$ 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 ${\cal N}$ has instances of nontrivial dominions, then for any proper subvariety ${\cal Q}$ of ${\cal G}roup$, ${\cal NQ}$ also has instances of nontrivial dominions.Comment: Plain TeX, 22 pp. Typos corrected. Final versio