Character Theory of finite groups1. $p$Parts of Character degrees
Let $G$ be a finite group and $p$ be a prime. We denote by
$\textrm{Irr}(G)$ the set of complex irreducible characters of $G$ and
by $\textrm{cd}(G)$ the character degrees of $G.$ The celebrated
It$\hat{\rm{o}}$Michler theorem states that $p$ does not divide $\chi (1)$ for
all $\chi \in \textrm{Irr}(G)$ if and only if $G$ has a normal abelian
Sylow $p$subgroup.
Many variations of this theorem have been proposed and studied in the literature. Recently, Lewis, Navarro and Wolf proved that if $G$ is a finite solvable group and for every $\chi \in \textrm{Irr}(G),$ $\chi (1)_p \le p$, then $G : \textrm{Fit} (G)_p \le p^2$ where $\textrm{Fit} (G)$ is the Fitting subgroup of $G$ and $m_p$ is the $p$part of $m\in \textbf{N}$. Furthermore, if $P$ is a Sylow $p$subgroup of $G,$ then $P'$ is subnormal in $G.$ For arbitrary groups, they proved that if every character $\chi \in \textrm{Irr}(G)$ satisfies $\chi (1)_2 \le 2$, then $G : \textrm{Fit}(G)_2 \le 2^3$ and $P''$ is subnormal in $G$ where $P$ is a Sylow $2$subgroup of $G$. The simple group $\textrm{A}_7$ shows that this bound is best possible. In the paper: Lewis, Mark L.; Navarro, Gabriel; Tiep, Pham Huu; TongViet, Hung P.;
$p$parts of character degrees. J. Lond. Math. Soc. (2) 92 (2015), no. 2, 483–497. we prove the following: Theorem. Let
$G$ be a finite group, and let $p$ be an odd prime. If
$\chi(1)_p\le p$ for all $\chi \in \textrm{Irr }(G)$, then $G:
\textbf{O}_p (G)_p \le p^{4}$. Moreover, if $P$ is a Sylow
$p$subgroup of $G$, then $P''$ is subnormal in $G$.
Notice that $G:\textrm{Fit}(G)_p=G:\textbf{O}_p(G)_p$ for a finite group $G$ and a prime $p.$ For $p$solvable groups, we obtained a stronger result. Theorem. Let
$p$ be an odd prime and let $G$ be a finite $p$solvable group.
If $\chi(1)_p\le p$ for all $\chi \in \textrm{Irr }(G)$, then
$G/\textrm{sol} (G)_p \leq p$ and $G/\textbf{O}_p (G)_p \leq p^3$.
Recall that $\textrm{sol} (G)$ is the solvable radical of $G.$ We suspect that the correct bound in both theorems is $G: \textbf{O}_p (G)_p \le p^{2}.$ Our study also suggests the following. Conjecture.
If $p^a$ is the largest power of $p$ dividing the degrees of
irreducible characters of $G$, then $G:\textbf{O}_p(G)_p\le p^{f(a)}$
where $f (a)$ is a function in $a$ and $P^{(a+1)}$ is subnormal in $G$.
Moreover, we conjecture that: \[f(a)= \left\{\begin{array}{cc}{2a},& \text{if
$p>2$}\\
{3a},&\text{if $p=2.$} \end{array}\right.\] Extending these results to block theory, we propose the following conjecture.
Conjecture. Let
$G$ be a finite group and let $p$ be a prime. Let $B$ be a block of $G$
with defect group $D.$ If $\chi(1)_p\le p$ for all
$\chi\in\textrm{Irr}(B),$ then $\mu(1)\le p$ for all $\mu\in
\textrm{Irr}(D).$
For $p$solvable groups, this conjecture can be reduced to the following. Conjecture.
Let $N$ be a normal subgroup of a finite group $G$ and let $p$ be a
prime. Let $\theta\in\textrm{Irr(N)}$ and let $P/N$ be a Sylow
$p$subgroup of $G/N.$ If $\chi(1)_p\le p$ for all
$\chi\in\textrm{Irr}(G)$ lying over $\theta,$ then $\mu(1)\le p$ for
all $\mu\in\textrm{Irr}(P/N).$
Clearly, this is an extension of GluckWolf theorem and is the main ingredient for the proof of Brauer's height zero conjecture in the $p$solvable case. The general case of this theorem was recently proved by G. Navarro and P.H. Tiep. We now turn to Brauer characters. Let $\textrm{IBr}_p (G)$ be the set of irreducible $p$Brauer characters of $G$ and $\textrm{cd}_p(G)$ the set of the $p$Brauer character degrees of $G$. There are some significant differences between ordinary character degrees and $p$Brauer character degrees; for example, the Brauer degrees need not divide the order of the group, and a Brauer character version of the It$\hat{\rm{o}}$Michler theorem only holds for the given prime $p$. That is, if $p$ divides no $p$Brauer degree of a finite group $G$, then $G$ has a normal Sylow $p$subgroup. Now FongSwan theorem implies that for a $p$solvable group $G,$ if $\chi(1)_p\le p$ for all $\chi\in\textrm{Irr}(G)$ then $\varphi(1)_p\le p$ for all $\varphi\in\textrm{IBr}_p(G).$ Moving away from $p$solvable groups, this condition does not hold. For example, if one takes $G = \textrm{M}_{22}$ and $p = 2$, then $\textbf{O}_2 (G) = 1$ and $\beta (1)_2 \le 2$ for all $\beta \in \textrm{IBr}_p (G)$ but $G_2 = 2^7$ and there exists a character $\chi \in \textrm{Irr} (G)$ with $\chi (1)_2 = 2^3$. However, if the group has an abelian Sylow $p$subgroup, then a recent result of Kessar and Malle on Brauer's height zero conjecture implies the following. Theorem. Let
$p$ be a prime and let $G$ be a finite group with $\textbf{O}_p (G) =
1$. If $G$ has an abelian Sylow $p$subgroup and $\varphi (1)_p
\le p$ for every $\varphi\in\textrm{IBr}_p(G)$, then $\chi (1)_p \le p$
for every $\chi \in \textrm{Irr} (G).$
As a corollary, we deduce that for a $p$solvable group $G$, $\chi (1)_p \le p$ for all $\chi \in \textrm{Irr }(G)$ if and only if $\varphi (1)_p \le p$ for all $\varphi \in \textrm{IBr} (G).$ Obviously, for arbitrary finite groups, we do not have such an equivalence. Nevertheless, we obtain the following. Theorem. Let
$p$ be a prime and $G$ be a finite group with $\textbf{O}_p (G) =
1$. If $\beta (1)_p \le p$ for all $\beta \in \textrm{IBr}_p
(G)$, then the following hold.
It seems that the bounds in the previous theorem are probably not best
possible. We conjecture that the correct bounds should be
$2^7$ in $(1),$ $p^2$ in $(2),$ and $3^3$ in (3).
2. $p$Parts of Brauer charactersIn
general, not much can be said about the degrees of $p$Brauer
characters of arbitrary finite groups. However, $p$Brauer
character degrees display a slightly better behavior if we
consider their $p$parts. For instance, a theorem of Michler asserts
that $\phi(1)_p=1$ for all $\phi \in \textrm{IBr}(G)$ (that is, $p$
does not divide $\phi(1)$ for all $\phi \in \textrm{IBr}(G)$) if
and only if $G$ has a normal Sylow $p$subgroup. Since ${\rm
cd}_p(G)={\rm cd}_p(G/\textbf{O}_p(G))$ (because
$\textbf{O}_p(G)$ is in the kernel of the irreducible $p$modular
representations), we see that when dealing with $p$Brauer character
degrees, we may generally assume that $p$ divides some $m \in
{\rm cd}_p(G)$.
In our paper: Navarro, Gabriel; Tiep, Pham Huu; TongViet, Hung P. $p$parts of Brauer character degrees. J. Algebra 403 (2014), 426–438.
we have been able to prove the following. Theorem. Let $G$ be a finite group and let $p$ be an odd prime. Suppose that the degrees of all nonlinear irreducible $p$Brauer characters of $G$ are divisible by $p$.
We note that if $G = \textrm{PSL}_2(27) \cdot 3$, then we have that
$\textrm{cd}_3(G) = \{1,9,12,27,36\}$, which shows that the theorem above fails
for $p = 3$ and that it is best possible. We should also mention that
under the conditions of this theorem, the prime $2$ behaves
somehow in the opposite way: it is often the case that all
nonlinear irreducible 2Brauer characters of nonsolvable groups have
even degree; in fact, the number of their 2parts can be quite large (with the exception of
$\textrm{M}_{22}$ where all nonlinear $2$Brauer character degrees
have the same 2part).
Theorem. Let $G$ be a finite group and let $p$ be an odd prime. Suppose that $\textrm{cd}_p(G)=\{1,m\}$ with $m>1.$ Then $G$ is solvable.
Since $\textrm{cd}_5(\textrm{A}_5)=\{1,3,5\}$, we see that this theorem cannot be further generalized. Observe that $\textrm{cd}_2(\textrm{PGL}_2(q))=\{1, q1\}$ whenever $q=9$ or a Fermat prime, so for $p=2$, there are nonsolvable groups satisfying the hypothesis of the previous theorem.
Theorem. Let $G$ be a nonsolvable group with $\textbf{O}_2(G)=1$. Then
$\textrm{cd}_2(G) = \{1,m\}$ with $m > 1$ if and only if the
following conditions hold:
