| DATE: | Thursday, September 12, 2002 |
| TIME: | 4:30-5:30 PM |
| PLACE: | LN 2205 |
| SPEAKER: | Luise-Charlotte Kappe (Binghamton University) |
| TITLE: | Capable groups and nonabelian tensor squares |
Finite simple groups have been classified some twenty years ago and so it seems all what remains to do the same for finite groups is to find out how the simple groups "stack up". In case of finite groups of p-power order, where the only building block is the cyclic group of order p, "stacking up" is the real problem. This was clearly recognized by Philip Hall in his paper on the classification of prime-power groups over sixty years ago. One of the questions Hall singled out is whether a group is capable, i.e. being the central quotient of another group.
In 1938, Reinhold Baer characterized the finitely generated abelian groups which are capable as those groups where the two factors of highest order in a direct composition have equal order. Up to now this remained the only classification of capable groups within a class of groups.
In 1979, Beyl, Felgner and Schmid characterized capable groups as those groups with trivial epicenter which is defined in terms of the central extensions of a group. But until Graham Ellis characterized the epicenter of a group via its nonabelian tensor square, there were hardly any methods to determine the epicenter. Using this result by Ellis, the epicenter of a group can be determined, once its nonabelian tensor square is given. This enabled us to characterize the capable groups among the 2-generator p-groups of nilpotency class 2 with p an odd prime.
Though a characterization of capable p-groups of nilpotency class 2 seems to be still not in reach, it should be added that Baer's criterion, namely that the two generators of highest order have equal order, which is necessary and sufficient for finitely generated abelian groups, is only necessary but not sufficient for finite p-groups of odd order and nilpotency class 2.
It should be added here that on our way to these results, we compute various homological functors for these groups and some invariants of topological interest, among them a curious result concerning the Isomorphism Theorem of Hurewicz.