Leírás
A finite p-group of nilpotency class 2 is said to be special if its centre, commutator subgroup and Frattini subgroup all coincide and is an elementary abelian p-group. An extra-special p-group is a special p-group with cyclic centre. The classical structure theorem states that
(a) every non-abelian special p-group is a subdirect product of groups which are the central product of an extra-special p-group and an abelian group;
(b) every extra-special p-group is the internal central product extra-special p-groups of order p^3;
(c) extra-special p-groups of order p^3 are classified: these are the two non-abelian groups of order p^3.
The talk will be about the following generalisation.
(A) Every finite(ly generated) 2-nilpotent group is a subdirect products of such groups with cyclic centre;
(B) every group from the latter family is isomorphic to the central product of 2-generated 2-nilpotent groups with cyclic centre;
(C) these 2-generated groups are classified.
As an application, we show that every finite(ly generated) 2-nilpotent group is a subgroup of a suitable 3x3 upper triangular matrix group that generalises the Heisenberg group. Further applications involve that every finite 2-nilpotent group of bounded rank act faithfully on a suitable variety birationally and on a suitable manifold via diffeomorphisms. Actually the structure theorem above emerged from the proof of this statement. By a recent result of Attila Guld, this application contains the largest possible family of finite groups (up to bounded extensions) whose members can all act birationally on the same variety.
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https://zoom.us/j/93592331759?pwd=emtRczUrZ1dObjg1UVc3VmhSU0hEZz09
Meeting ID: 935 9233 1759
Passcode: 511252