# What conditions guarantee that a group is in the following collection?

This is a follow-up of a sort to a question I asked earlier, but this turned out to be what was really interesting, and the main relation between the questions is that they share some definitions, which I will mention again here anyway, and that the end result is the Taketa inequality.

First some definitions: let $cd(G)$ be the set of degrees of irreducible complex characters of the group $G$. If these degree are $1 = f_1 < f_2 < \dots < f_t$ and $\chi$ is a character of $G$ with $\psi$ an irreducible constituent of $\chi$ of lowest possible degree and $\varphi$ an irreducible constituent of $\chi$ of largest possible degree, such that $\psi(1) = f_n$ and $\varphi(1) = f_m$, then define $s(\chi) = n$ and $v(\chi) = m$. Let $G$ be a finite group, let $\chi$ be an irreducible character of $G$ of degree $f_k$ and let $H$ be a subgroup of $G$. We call $(G,H,\chi)$ a good triple if there is an irreducible character $\psi$ of $H$ such that $s(\chi_H) + v(\psi^G) \leq k$.

Now define the collection of finite group $X$ by the property that a finite group $G$ is in $X$ if and only if either $G$ is an $M$-group, or if for each non-linear irreducible character $\chi$ of $G$, there is a proper subgroup $H$ of $G$ such that $H\in X$ and $(G,H,\chi)$ is a good triple.

I am interested in what "nice" properties of a group will guarantee that it is in this collection. I have shown that nilpotent groups and groups of squarefree order are in $X$, but I would like to find some groups in $X$ which are not $M$-groups, since I can prove that the Taketa inequality holds for all groups in $X$, and this is of course not really interesting unless $X$ contains groups other than $M$-groups

Edit: Changed the definition a bit to include all $M$-groups, since I found out that I can still prove the same things about these groups, and the collection becomes considerably larger this way.

Edit2: I have found that knowing the character degrees of the derived subgroup is often enough to determine the existence of good triples, namely via the following result:

If $\chi\in Irr(G)$ with $\chi(1) = f_k$ (with $f_k$ as above) then define $pos(\chi) = k$ (the position of $\chi$). If $\chi$ is not linear and $(G,G',\chi)$ is not a good triple, then there is a character $\psi\in Irr(G')$ such that $pos(\psi)\geq pos(\chi)$ and $\psi(1)$ divides $\chi(1)$ (actually, such that $\psi(1)t$ divides $\chi(1)$ where $t = |G:I_G(\psi)|$)

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Do you have a favorite reference to the Taketa inequality you might be able to link to? – GS May 11 '10 at 11:14
The Taketa inequality is $dl(G) \leq |cd(G)|$. Taketa proved that this holds for M-groups (ie, where all irreducible complex characters are induced from linear characters of subgroups). It is conjectured to hold for all solvable groups, and there are many positive results in that direction (it has been proven when $|cd(G)|\leq 5$ and when $|G|$ is odd). – Tobias Kildetoft May 12 '10 at 11:42
I have now run some calculations in GAP, and the smallest group in the collection which is not an $M$-group has order 96. There are two such of order 96, namely $(SL(2,3)\times C_2)\rtimes C_2$ and $(SL(2,3)\rtimes C_2)\rtimes C_2$. I calculated up to order 250 and found that there are 20 groups in the collection which are not $M$-groups up to that order, so they are fairly sparse. I was not able to compute for larger orders as I kept running out of memory. – Tobias Kildetoft May 27 '10 at 13:35