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

Question

[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?

Answer 1

A good potential supply of this sort of character is provided by Frobenius groups. These are finite groups $G$ of the form $G = KH$, where $K \cap H = 1$ and $K \lhd G$, and furthermore $C_{G}(x) \leq K$ for all non-identity elements $x \in K$. By a Theorem of Thompson the group $K$ is necessarily nilpotent.

One Frobenius group not mentioned in your examples is given by $H \cong {\rm SL}(2,5)$ and $K$ elementary Abelian of order $121$, which admits a regular action (on non-identity elements) by $H$. The semidirect product gives an example of a finite group $G$ which is not solvable, but has an irreducible character of quite large degree. Frobenius complements which are not solvable are very rare though.

In general, the irreducible characters $\chi$ of a (general) Frobenius group $G$ which do not contain $K$ in their kernels have degrees of the form $|H|\mu(1)$, where $\mu$ is an irreducible character of $K$. For such a $\chi$ to have quite large degree, we clearly need $|H|\mu(1)^2 \geq 0.64 |K|$. Notice also that $|K| \equiv 1$ (mod $|H|$). If $|H| < |K|-1$, then we obtain the contradiction $\mu(1)^2 > 1.28 |K|$. Thus the only Frobenius groups with irreducible characters of quite large degree are those in which the complement $H$ acts transitively on non-identity elements of the kernel $K$. This forces the kernel $K$ to be elementary Abelian. Anyway, it looks as though you won't get many new examples from Frobenius groups (later edit-as is made explicit by Noah Snyder's comment below. Previous version did not say what I meant in any case).

Answer 2

There's an example on page 8 of my paper which can give you a group of size $q^3(q-1)$ and an irrep of dimension $q(q-1)$ for any prime power $q$. This gives a factor of $\sqrt{\frac{q}{q-1}}$.

In my paper and Durfee-Jensen (there's also a preprint of Isaacs and a preprint of Larsen-Malle-Tiep which were written between those two papers), we're looking at representations whose dimensions are much much closer to $\sqrt{|G|}$. So those results can't be directly applied to your more general question. Nonetheless, the "moral" result of those papers is that "having a large character should mean you're in the case studied by Gagola and Kuisch-van der Waall." One could certainly hope that this result is true for a weaker definition of "large character" than we used. However, you should expect that proving this for a weaker definition of "large character" is going to get harder.

Answer 3

The following is a list of all groups whose order is $\leq 1000$ and not equal to one of $384, 768, 864, 896$ or $960$ which satisfy the condition in the title of the question -- for each group we give its GAP ID number, the largest character degree, its quotient by the square root of the group order and a structure description of the group:

• $[ 1, 1 ]$, $1$, $1.00000$, $1$

• $[ 6, 1 ]$, $2$, $0.81650$, ${\rm S}_3$

• $[ 12, 3 ]$, $3$, $0.86603$, ${\rm A}_4$

• $[ 20, 3 ]$, $4$, $0.89443$, ${\rm C}_5 \rtimes {\rm C}_4$

• $[ 42, 1 ]$, $6$, $0.92582$, $({\rm C}_7 \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 54, 5 ]$, $6$, $0.81650$, $(({\rm C}_3 \times {\rm C}_3) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 54, 6 ]$, $6$, $0.81650$, $({\rm C}_9 \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 56, 11 ]$, $7$, $0.93541$, $({\rm C}_2 \times {\rm C}_2 \times {\rm C}_2) \rtimes {\rm C}_7$

• $[ 72, 39 ]$, $8$, $0.94281$, $({\rm C}_3 \times {\rm C}_3) \rtimes {\rm C}_8$

• $[ 72, 41 ]$, $8$, $0.94281$, $({\rm C}_3 \times {\rm C}_3) \rtimes {\rm Q}_8$

• $[ 110, 1 ]$, $10$, $0.95346$, $({\rm C}_{11} \rtimes {\rm C}_5) \rtimes {\rm C}_2$

• $[ 156, 7 ]$, $12$, $0.96077$, $({\rm C}_{13} \rtimes {\rm C}_4) \rtimes {\rm C}_3$

• $[ 192, 184 ]$, $12$, $0.86603$, $(({\rm C}_2 \times {\rm C}_2 \times {\rm C}_2 \times {\rm C}_2) \rtimes {\rm C}_3) \rtimes {\rm C}_4$

• $[ 192, 185 ]$, $12$, $0.86603$, $(({\rm C}_4 \times {\rm C}_4) \rtimes {\rm C}_3) \rtimes {\rm C}_4$

• $[ 192, 1008 ]$, $12$, $0.86603$, $((({\rm C}_4 \times {\rm C}_4) \rtimes {\rm C}_3) \rtimes {\rm C}_2) \rtimes {\rm C}_2$

• $[ 192, 1009 ]$, $12$, $0.86603$, $((({\rm C}_2 \times {\rm C}_2 \times {\rm C}_2 \times {\rm C}_2) \rtimes {\rm C}_3) \rtimes {\rm C}_2) \rtimes {\rm C}_2$

• $[ 192, 1023 ]$, $12$, $0.86603$, $((({\rm C}_4 \times {\rm C}_4) \rtimes {\rm C}_2) \rtimes {\rm C}_2) \rtimes {\rm C}_3$

• $[ 192, 1025 ]$, $12$, $0.86603$, $(({\rm C}_2 \times {\rm C}_2) . ({\rm C}_2 \times {\rm C}_2 \times {\rm C}_2 \times {\rm C}_2)) \rtimes {\rm C}_3$

• $[ 240, 191 ]$, $15$, $0.96824$, $(({\rm C}_2 \times {\rm C}_2 \times {\rm C}_2 \times {\rm C}_2) \rtimes {\rm C}_5) \rtimes {\rm C}_3$

• $[ 272, 50 ]$, $16$, $0.97014$, ${\rm C}_{17} \rtimes {\rm C}_{16}$

• $[ 342, 7 ]$, $18$, $0.97333$, $({\rm C}_{19} \rtimes {\rm C}_9) \rtimes {\rm C}_2$

• $[ 486, 31 ]$, $18$, $0.81650$, $({\rm C}_{27} \rtimes {\rm C}_9) \rtimes {\rm C}_2$

• $[ 486, 39 ]$, $18$, $0.81650$, $(({\rm C}_9 \rtimes {\rm C}_9) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 486, 40 ]$, $18$, $0.81650$, $({\rm C}_3 . (({\rm C}_9 \times {\rm C}_3) \rtimes {\rm C}_3)) \rtimes {\rm C}_2$

• $[ 486, 41 ]$, $18$, $0.81650$, $(({\rm C}_9 \rtimes {\rm C}_9) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 486, 127 ]$, $18$, $0.81650$, $(({\rm C}_3 \times ({\rm C}_9 \rtimes {\rm C}_3)) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 486, 129 ]$, $18$, $0.81650$, $(({\rm C}_3 \times ({\rm C}_9 \rtimes {\rm C}_3)) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 486, 131 ]$, $18$, $0.81650$, $(({\rm C}_3 \times (({\rm C}_3 \times {\rm C}_3) \rtimes {\rm C}_3)) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 486, 176 ]$, $18$, $0.81650$, $(({\rm C}_3 \times ({\rm C}_9 \rtimes {\rm C}_3)) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 486, 177 ]$, $18$, $0.81650$, $(({\rm C}_3 \times ({\rm C}_9 \rtimes {\rm C}_3)) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 486, 178 ]$, $18$, $0.81650$, $(({\rm C}_3 \times (({\rm C}_3 \times {\rm C}_3) \rtimes {\rm C}_3)) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 486, 179 ]$, $18$, $0.81650$, $(({\rm C}_3 . (({\rm C}_3 \times {\rm C}_3) \rtimes {\rm C}_3)) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 486, 236 ]$, $18$, $0.81650$, $(({\rm C}_3 \times (({\rm C}_3 \times {\rm C}_3) \rtimes {\rm C}_3)) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 486, 238 ]$, $18$, $0.81650$, $(({\rm C}_3 \times ({\rm C}_9 \rtimes {\rm C}_3)) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 500, 17 ]$, $20$, $0.89443$, $(({\rm C}_5 \times {\rm C}_5) \rtimes {\rm C}_5) \rtimes {\rm C}_4$

• $[ 500, 21 ]$, $20$, $0.89443$, $(({\rm C}_5 \times {\rm C}_5) \rtimes {\rm C}_5) \rtimes {\rm C}_4$

• $[ 500, 18 ]$, $20$, $0.89443$, $({\rm C}_{25} \rtimes {\rm C}_5) \rtimes {\rm C}_4$

• $[ 504, 160 ]$, $18$, $0.80178$, $(({\rm C}_7 \rtimes {\rm C}_3) \times {\rm A}_4) \rtimes {\rm C}_2$

• $[ 504, 161 ]$, $18$, $0.80178$, ${\rm A}_4 \times (({\rm C}_7 \rtimes {\rm C}_3) \rtimes {\rm C}_2)$

• $[ 506, 1 ]$, $22$, $0.97802$, $({\rm C}_{23} \rtimes {\rm C}_{11}) \rtimes {\rm C}_2$

• $[ 600, 148 ]$, $24$, $0.97980$, $(({\rm C}_5 \times {\rm C}_5) \rtimes {\rm C}_3) \rtimes {\rm C}_8$

• $[ 600, 149 ]$, $24$, $0.97980$, $(({\rm C}_5 \times {\rm C}_5) \rtimes {\rm C}_8) \rtimes {\rm C}_3$

• $[ 600, 150 ]$, $24$, $0.97980$, $(({\rm C}_5 \times {\rm C}_5) \rtimes {\rm Q}_8) \rtimes {\rm C}_3$

• $[ 672, 1257 ]$, $21$, $0.81009$, $({\rm C}_2 \times {\rm C}_2 \times (({\rm C}_2 \times {\rm C}_2 \times {\rm C}_2) \rtimes {\rm C}_7)) \rtimes {\rm C}_3$

• $[ 672, 1258 ]$, $21$, $0.81009$, $(({\rm C}_2 \times {\rm C}_2 \times {\rm C}_2) \rtimes {\rm C}_7) \times {\rm A}_4$

• $[ 702, 47 ]$, $26$, $0.98131$, $(({\rm C}_3 \times {\rm C}_3 \times {\rm C}_3) \rtimes {\rm C}_{13}) \rtimes {\rm C}_2$

• $[ 812, 7 ]$, $28$, $0.98261$, $({\rm C}_{29} \rtimes {\rm C}_7) \rtimes {\rm C}_4$

• $[ 840, 139 ]$, $24$, $0.82808$, $({\rm C}_5 \rtimes {\rm C}_4) \times (({\rm C}_7 \rtimes {\rm C}_3) \rtimes {\rm C}_2)$

• $[ 930, 1 ]$, $30$, $0.98374$, $(({\rm C}_{31} \rtimes {\rm C}_5) \rtimes {\rm C}_3) \rtimes {\rm C}_2$

• $[ 992, 194 ]$, $31$, $0.98425$, $({\rm C}_2 \times {\rm C}_2 \times {\rm C}_2 \times {\rm C}_2 \times {\rm C}_2) \rtimes {\rm C}_{31}$