By Editors: Campbell, Robertson and Smith
ISBN-10: 0511065612
ISBN-13: 9780511065613
ISBN-10: 0521537401
ISBN-13: 9780521537407
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Additional info for Groups St Andrews 2001 in Oxford, Vol 2
Example text
R(G, V ) (where r(G, V ) is as in Section 2), are representatives of the orbits 322 KELLER of G on V , then r(G,V ) (1) k(CG (vi )) = k(GV ) = i=1 1 |G| |CG (v)| k(CG (v)). v∈V Alternatively, if gi , i = 1, . . , k(G) are representatives of the conjugacy classes of G, then k(G) (2) r(CG (gi ), CV (gi )) = k(GV ) = i=1 1 |G| |CG (g)| r(CG (g), CV (g)). g∈G These elegant formulas look meaningful and seem to be a key to the solution of the k(GV )–problem, but surprisingly in the development up to this point they have hardly been used at all, except for settling easy cases such as G abelian (see [109]) or more recently in [36] to solve the case p = 31.
The point of this is that the existence of weakly real vectors is strong enough to yield the wanted conclusion k(GV ) ≤ |V |, but it is still weak enough to be true for all GV at least if p is sufficiently large. And its key property is that unlike the property k(GV ) ≤ |V | it has a good inductive behaviour which makes the inductive proof possible. So it is just about the “right” property to look at. Other properties like the existence of regular orbits, which also imply k(GV ) ≤ |V |, are too strong and even for large p simply are not always satisfied, and correspondingly their inductive behaviour is not good enough.
All these results indicate that dl(G) might grow slower than linearly with |cd(G)|. As it happens, the results on orbit sizes presented above shed some more light on the question. Namely if V is a G–module, then so is V0 = Irr(V ), and we consider the semidirect product GV0 . 18)] every λ ∈ V0 can be extended to its inertia group in GV , from which it can be induced to an irreducible character of GV . This shows that every orbit size of G on V0 is a complex irreducible character degree of GV . Using this allows a translation of results on orbit sizes to results on character degrees.
Groups St Andrews 2001 in Oxford, Vol 2 by Editors: Campbell, Robertson and Smith
by David
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