Classifying finite p-groups by coclass

Host Institution:

La Trobe University

Title of Seminar:

Classifying finite p-groups by coclass

Speaker's Name:

Dr Heiko Dietrich

Speaker's Institution:

Monash University

Time and Date:

Friday 8 November, 2.00pm (AEDT)

Seminar Abstract:

One of the major themes in group theory is the classification of groups up to isomorphism. A famous example is the 'Classification Theorem of Finite Simple Groups', which classifies all finite simple groups - the basic building blocks of all finite groups. However, even if we restrict attention to the least complicated finite simple group, the cyclic group of prime order p, it is still an intricate problem to put these groups together in order to construct all groups of p-power order, so-called finite p-groups, up to isomorphism.

One approach for classifying finite p-groups is to fix the order, say p^n, and to classify all groups having this order. However, already for small n this problem becomes intractable. In this talk, we discuss an alternative approach, namely, a classification of p-groups by "coclass". If the coclass is fixed, say r, then the finite p-groups of coclass r (up to isomorphism) define an infinite graph, the so-called coclass graph G(p,r). This graph has some remarkable periodic structures, which are reflected in the structures of the groups. We will discuss the structure of this graph, some classification results, and some open conjectures.

 

 

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