2,652 research outputs found
Fixed conjugacy classes of normal subgroups and the k(GV)-problem
We establish several new bounds for the number of conjugacy classes of a
finite group, all of which involve the maximal number c of conjugacy classes of
a normal subgroup fixed by some element of a suitable subset of the group. To
apply these formulas effectively, the parameter c, which in general is hard to
control, is studied in some important situations. These results are then used
to provide a new, shorter proof of the most difficult case of the well-known
k(GV)-problem, which occurs for p=5 and V induced from the natural module of a
5-complement of GL(2,5). We also show how, for large p, the new results reduce
the k(GV)-problem to the primitive case, thereby improving previous work on
this. Furthermore, we discuss how they can be used in tackling the imprimitive
case of the as of yet unsolved noncoprime k(GV)-problem
Derived length and conjugacy class sizes
AbstractLet G be a finite solvable group, and let F(G) be its Fitting subgroup. We prove that there is a universal bound for the derived length of G/F(G) in terms of the number of distinct conjugacy class sizes of G. This result is asymptotically best possible. It is based on the following result on orbit sizes in finite linear group actions: If G is a finite solvable group and V a finite faithful irreducible G-module of characteristic r, then there is a universal logarithmic bound for the derived length of G in terms of the number of distinct r′-parts of the orbit sizes of G on V. This is a refinement of the author's previous work on orbit sizes
A new lower bound for the number of conjugacy classes
In 2003, H\'{e}thelyi and K\"{u}lshammer proposed that if is a finite
group and is a prime dividing the group order, then ,
and they proved this conjecture for solvable and showed that it is sharp
for those primes for which is an integer. This initiated a
flurry of activity, leading to many generalizations and variations of the
result; in particular, today the conjecture is known to be true for all finite
groups. In this note, we put forward a natural new and stronger conjecture,
which is sharp for all primes , and we prove it for solvable groups, and
when is large, also for arbitrary groups.Comment: A hypothesis for Theorem E is added that was inadvertently omitted in
the previous version. Some other small changes have been mad
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