2 corrected some mistakes

[edited in response to some corrections by Geoff Robinson and F. Ladisch]

Throughout, all my groups are finite, and all my representations are over the complex numbers.

If $G$ is a group and $\chi$ is an irrreducible character on it, of degree $d$, say that $\chi$ has quite large degree if $d\geq (4/5)|G|^{1/2}$.

This condition has arisen in some work I am doing concerning (Banach algebras built on) restricted direct products of finite groups, and while our main concern was to find some examples with this property, I am curious to know if anything more can be said.

(I've had a quick look at the papers of Snyder (Proc AMS, 2008) and Durfee & Jensen (J. Alg, 2011), but these don't seem to quite give what I'm after. However, I may well have missed something in their remarks.)

Examples. The affine group over a finite field with $q$ elements has order $q(q-1)$ and a character of degree $q-1$, which has "quite large degree" for $q\geq 3$. The affine group of the ring ${\mathbb Z}/p^n{\mathbb Z}$ has order $p^{2n-1}(p-1)$ and a character of degree $p^{n-1}(p-1)$, which has quite large degree for $p\geq 3$. Being fairly inexperienced in the world of finite groups, I can't think of other examples (except by taking finite products of some of these examples).

Some simple-minded observations from a bear of little brain. A group can have at most one character of quite large degree (this is immediate from $|G|=\sum_\pi d_\pi^2$). If such a character $\chi$ exists, it is real (hence rational) valued (edit: as pointed out by Geoff Robinson in comments, $\chi$ must be equal to its Galois conjugates, and hence rational valued), and its centre is trivial (consider its $\ell^2$-norm). In particular $G$ can't be nilpotent. Moreover, since $\chi\phi=\chi$ for any linear character $\phi$, a bit of thought shows that $\chi$ vanishes outside the derived subgroup of $G$.

Now, let ${\mathcal C}$ be the class of all groups $G$ that possess a character of quite large degree ; clearly this is closed under taking finite products. Let ${\mathcal S}$ denote the class of all solvable groups.

Question 1. Does there exist a finite collection ${\mathcal F}$ of groups such that every group in ${\mathcal C}$ is the product of groups in ${\mathcal F}\cup{\mathcal S}$?

Question 2. Can we bound the derived length of groups in ${\mathcal C}\cap{\mathcal S}$?

Question 3. If the answers to Q1 and Q2 are negative, or beyond current technology, would anything improve if we replaced $4/5$ with $1-\epsilon$ for one's favourite small $\epsilon$?

Question 4. It may well be the case that hoping for a description of the class ${\mathcal C}$ in terms of familiar kinds of group is far too naive and optimistic. If so, could someone please give me some indications as to why? Perhaps it reduces to a set of known open problems?

1

# Groups of order $n$ with a character whose degree is at least $0.8\sqrt{n}$ (say)

Throughout, all my groups are finite, and all my representations are over the complex numbers.

If $G$ is a group and $\chi$ is an irrreducible character on it, of degree $d$, say that $\chi$ has quite large degree if $d\geq (4/5)|G|^{1/2}$.

This condition has arisen in some work I am doing concerning (Banach algebras built on) restricted direct products of finite groups, and while our main concern was to find some examples with this property, I am curious to know if anything more can be said.

(I've had a quick look at the papers of Snyder (Proc AMS, 2008) and Durfee & Jensen (J. Alg, 2011), but these don't seem to quite give what I'm after. However, I may well have missed something in their remarks.)

Examples. The affine group over a finite field with $q$ elements has order $q(q-1)$ and a character of degree $q-1$, which has "quite large degree" for $q\geq 3$. The affine group of the ring ${\mathbb Z}/p^n{\mathbb Z}$ has order $p^{2n-1}(p-1)$ and a character of degree $p^{n-1}(p-1)$, which has quite large degree for $p\geq 3$. Being fairly inexperienced in the world of finite groups, I can't think of other examples (except by taking finite products of these examples).

Some simple-minded observations from a bear of little brain. A group can have at most one character of quite large degree (this is immediate from $|G|=\sum_\pi d_\pi^2$). If such a character $\chi$ exists, it is real (hence rational) valued, and its centre is trivial (consider its $\ell^2$-norm). In particular $G$ can't be nilpotent. Moreover, since $\chi\phi=\chi$ for any linear character $\phi$, a bit of thought shows that $\chi$ vanishes outside the derived subgroup of $G$.

Now, let ${\mathcal C}$ be the class of all groups $G$ that possess a character of quite large degree; clearly this is closed under taking finite products. Let ${\mathcal S}$ denote the class of all solvable groups.

Question 1. Does there exist a finite collection ${\mathcal F}$ of groups such that every group in ${\mathcal C}$ is the product of groups in ${\mathcal F}\cup{\mathcal S}$?

Question 2. Can we bound the derived length of groups in ${\mathcal C}\cap{\mathcal S}$?

Question 3. If the answers to Q1 and Q2 are negative, or beyond current technology, would anything improve if we replaced $4/5$ with $1-\epsilon$ for one's favourite small $\epsilon$?

Question 4. It may well be the case that hoping for a description of the class ${\mathcal C}$ in terms of familiar kinds of group is far too naive and optimistic. If so, could someone please give me some indications as to why? Perhaps it reduces to a set of known open problems?