Orbits of the actions of finite solvable groups

Abstract

2016-11-15T19:40:59

Blocks of small defect

“…In [25], Yang strengthened this result by showing the following. Suppose that V is a faithful completely reducible G-module where G is a finite solvable group, then there exist v ∈ V and K G such that C G (v) ⊆ K, where the Fitting length of K is less than or equal to 7.…”

“…Suppose that V is a faithful completely reducible G-module where G is a finite solvable group, then there exist v ∈ V and K G such that C G (v) ⊆ K, where the Fitting length of K is less than or equal to 7. An example [25,Section 4] was provided to show that the improvement is the best possible. Although one cannot say more in general because of this example, it is possible to show that there exits an element v ∈ V such that the p-part of C G (v) is relatively small for all the primes p ≥ 5.…”

Regular orbits of finite primitive solvable groups

“…We also construct groups with no regular orbits on V when e = 8, 9 and 16. When one studies the orbit structure of solvable linear groups, the common strategy is to use reduction. As we can see in many places (for example [1,6,7,9]), many problems are reduced to the situation where the action is irreducible and quasi-primitive and then properties are checked for all possible values of e. Our result is quite useful since the existence of regular orbit is a nice property which will imply many other properties. Clearly when e = 1, it is easy to find primitive solvable group which has no regular orbits on the vector space.…”

“…A/F ∈ SCRSp(2, 2) × SCRSp (6,3) and |A/F | 6 · 24 3 · 6 by Lemma 2.17(1), (9). By Lemma 2.3, |G| 6 · 24 3 · 6 · 54 2 · 6 < |W | 13 , a contradiction.…”

Bounding an Index by the Largest Character Degree of ap-Solvable Group

“…In particular, they showed that G F G p ≤ c p G 19 when G is any solvable group and that G F G p ≤ c p G 2 when G is odd. More recently, the work of Yang in Corollary 5.6 of [14] can be used to improve this bound to G F G p ≤ c p G 15 . An example in [4] shows that the bound is best possible when G is odd.…”