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Arturo Magidin

Publications and source records attributed to Arturo Magidin.

At least 19 recordsLinked to original sources
arXiv2009

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.

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arXiv2008

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.

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arXiv2006

Capability of nilpotent products of cyclic groups II

In Part I it was shown that if G is a p-group of class k, generated by elements of orders 1 1 and alpha_r <= alpha_{r-1} + [(k-1)/(p-1)]. It was also shown that when G is the k-nilpotent product of the cyclic groups generated by those elements and k=p=2 or k<p, then the given conditions are also sufficient. We make a correction related to the small class case, and extend the sufficiency result to k=p for arbitrary prime p.

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arXiv2006

Capability of nilpotent products of cyclic groups

A group is called capable if it is a central factor group. We consider the capability of nilpotent products of cyclic groups, and obtain a generalization of a theorem of Baer for the metabelian small class case. The approach is also used to obtain some recent results on the capability of certain nilpotent groups of class 2. We also prove a necessary condition for the capability of an arbitrary p-group of class k, and some further results.

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arXiv2005

Capable groups of prime exponent and class 2, II

We consider the capability of $p$ groups of class two and odd prime exponent. We use linear algebra and counting arguments to establish a number of new results. In particular, we settle the 4-generator case, and prove a sufficient condition based on the ranks of $G/Z(G)$ and $[G,G]$.

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arXiv2004

On the orders of generators of capable $p$-groups

A group is called capable if it is a central factor group. For each prime $p$ and positive integer $c$, we prove the existence of a capable $p$-group of class $c$ minimally generated by an element of order $p$ and an element of order $p^{1+\lfloor\frac{c-1}{p-1}\rfloor}$. This is best possible.

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arXiv2004

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.

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arXiv2003

Capability of certain nilpotent products of cyclic groups

A group is called capable if it is a central factor group. We consider the capability of certain nilpotent products of cyclic groups, and obtain a generalisation of a theorem of Baer for the small class case. The approach may also be used to obtain some recent results on the capability of certain nilpotent groups of class 2. We also obtain a necessary condition for the capability of an arbitrary $p$-group of class $k$, and some further results.

math.GR↗
arXiv2001

Amalgams of nilpotent groups of class two

We give necessary and sufficient conditions for weak and strong embeddability of amalgams in each subvariety of the category of all nilpotent groups of class at most two; this generalizes B. Maier's result for the latter class. We also discuss dominions (in the sense of Isbell), and characterize the weak, strong, and special amalgamation bases for each subvariety, contrasting the resulting classes with one another.

math.GR↗
arXiv2001

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.

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arXiv2000

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.

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arXiv1999

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 $p$.

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arXiv1999

Dominions in varieties of nilpotent groups

We investigate the concept of dominion (in the sense of Isbell) in several varieties of nilpotent groups. We obtain a full description of dominions in the variety of nilpotent groups of class at most two. Then we look at the behavior of dominions of subgroups of groups in ${\cal N}_2$ when taken in the context of ${\cal N}_c$ with $c>2$. Finally we establish the existence of nontrivial dominions in the category of all nilpotent groups.

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arXiv1999

Dominions in finitely generated nilpotent groups

In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of a finitely generated nilpotent group of class two is trivial in the category of all metabelian nilpotent groups.

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arXiv1999

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.

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