Abstract
There is no possibility of finding a finite set of invariants that will define every p-group up to an isomorphism in a useful way. The coclass conjectures can be regarded as offering a way around this difficulty. The solution offered in these conjectures is to give a classification that is weaker than an isomorphism by taking a suitable invariant of a p-group; in this instance the coclass defined as n — c where the group has order p n and nilpotency class c; and seeking to classify p-groups by coclass. By this we mean that we prove a theorem that states that if G is any p-group of coclass r then G has a normal subgroup N, of order bounded by a function of p and r, such that G/N has a certain well-defined structure. This falls short of a classification up to isomorphism first by ignoring the subgroup N, and secondly by failing to classify precisely the groups G/N. We do however have a recipe for constructing the groups G/N which describes their structure closely and which can be followed through in detail in simple cases.
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Leedham-Green, C.R., McKay, S. (2000). On the Classification of p-groups and pro-p Groups. In: du Sautoy, M., Segal, D., Shalev, A. (eds) New Horizons in pro-p Groups. Progress in Mathematics, vol 184. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1380-2_2
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DOI: https://doi.org/10.1007/978-1-4612-1380-2_2
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