Some sufficient conditions for the Taketa inequality (original) (raw)
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A note on codegrees and Taketa's inequality
2021
Let G be a finite group and cd(G) will be the set of the degrees of the complex irreducible characters of G. Also let cod(G) be the set of codegrees of the irreducible characters of G. The Taketa problem conjectures if G is solvable, then dl(G) ≤ |cd(G)|, where dl(G) is the derived length of G. In this note, we show that dl(G) ≤ |cod(G)| in some cases and we conjecture that this inequality holds if G is a finite solvable group.
Taketa’s theorem for some character degree sets
Archiv der Mathematik, 2013
Let cd(G) be the set of irreducible complex character degrees of a finite group G. The Taketa problem conjectures that if G is a finite solvable group, then dl(G) |cd(G)|, where dl(G) is the derived length of G. In this note, we show that this inequality holds if either all nonlinear irreducible characters of G have even degrees or all irreducible character degrees are odd. Also, we prove that this inequality holds if all irreducible character degrees have exactly the same prime divisors. Finally, Isaacs and Knutson have conjectured that the Taketa problem might be true in a more general setting. In particular, they conjecture that the inequality dl(N) |cd(G | N)| holds for all normal solvable subgroups N of a group G. We show that this conjecture holds if cd(G | N) is a set of non-trivial p-powers for some fixed prime p.
New Trends in Characterization of Solvable Groups
webdoc.sub.gwdg.de
Abstract. We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for ...
On the Taketa bound for normally monomial p-groups of maximal class
Journal of Algebra, 2004
A longstanding problem in the representation theory of finite solvable groups, sometimes called the Taketa problem, is to find strong bounds for the derived length dl(G) in terms of the number |cd(G)| of irreducible character degrees of the group G. For p-groups an old result of Taketa implies that dl(G) ≤ |cd(G)|, and while it is conjectured that the true bound is much smaller (in fact, logarithmic) for large dl(G), it turns out to be extremely difficult to improve on Taketa's bound at all. Here, therefore, we suggest to first study the problem for a restricted class of p-groups, namely normally monomial p-groups of maximal class. We exhibit some structural features of these groups and show that if G is such a group, then dl(G) ≤ 1 2 |cd(G)| + 11 2 .
Taketa's theorem for relative character degrees
Rocky Mountain Journal of Mathematics, 2013
It has been conjectured by Isaacs that, for finite group G, the inequality dl (N) ≤ |cd (G | N)| holds for all normal solvable subgroups N of G. We show that this conjecture holds for M-groups. Also, we prove that, if G is solvable and the common-divisor graph Γ(G|N) is disconnected, then dl (N) ≤ |cd (G | N)|, which is a generalization of [5, Theorem A]. 2010 AMS Mathematics subject classification. Primary 20C15.
A Note on the Solvablity of Groups
Let M be a maximal subgroup of a finite group G and K/L be a chief factor such that L ≤ M while K M. We call the group M ∩ K/L a c-section of M. And we define Sec(M) to be the abstract group that is isomorphic to a c-section of M. For every maximal subgroup M of G, assume that Sec(M) is supersolvable. Then any composition factor of G is isomorphic to L 2 (p) or Z q , where p and q are primes, and p ≡ ±1(mod 8). This result answer a question posed by ref. [12].
Journal of Group Theory, 2007
A finite group G is said to be a PST-group if, for subgroups H and K of G with H Sylow-permutable in K and K Sylow-permutable in G, it is always the case that H is Sylowpermutable in G. A group G is a T *-group if, for subgroups H and K of G with H normal in K and K normal in G, it is always the case that H is Sylow-permutable in G. In this paper, we show that finite PST-groups and finite T *-groups are one and the same. A new characterisation of soluble PST-groups is also presented.
A note on the solvability of groups
Journal of Algebra, 2006
Let M be a maximal subgroup of a finite group G and K/L be a chief factor such that L ≤ M while K M. We call the group M ∩ K/L a c-section of M. And we define Sec(M) to be the abstract group that is isomorphic to a c-section of M. For every maximal subgroup M of G, assume that Sec(M) is supersolvable. Then any composition factor of G is isomorphic to L 2 (p) or Z q , where p and q are primes, and p ≡ ±1(mod 8). This result answer a question posed by ref. [12].