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Products of finite nilpotent groups

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Suppose A and B are subgroups of a group G. We say that G is the product of A and B if G=AB={ab : aA, bB}. A natural question to ask is whether properties of G can be deduced from properties of A and B. There is an extensive literature on this question. Many properties have been considered- see for example the book of Amberg, Franciosi and de Giovanni and that of Ballester-Bolinches, Esteban-Romero and Asaad.

Many results concentrate on the case of A and B nilpotent. Most results are aimed at restricting the structure of non-nilpotent products G; for example, under appropriate restrictions, G will be supersoluble. However very little is known about the structure when G is itself nilpotent.

If G is nilpotent, there are many invariants we could consider: derived length, class, coclass, breadth and rank as examples. Very little is known about any of these. I will describe what is known.

This talk is part of the Algebra Seminar series.

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