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; Tong-Viet, 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: \[f(a)= \left\{\begin{array}{cc}{2a},& \text{if $p>2$}\\
{3a},&\text{if $p=2.$}
\end{array}\right.\]

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 Fong-Swan 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.

1. If $p = 2$, then $|G|_2 \le 2^9.$
2. If $p \ge 5$ or if $p = 3$ and $\textrm{A}_7$ is not involved in $G$, then $|G|_p \le p^{4}.$
3. If $p = 3$ and $\textrm{A}_7$ is involved in $G$, then $|G|_3 \le 3^{5}.$

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).

In 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; Tong-Viet, 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 non-linear irreducible 2-Brauer characters of non-solvable groups have even degree; in fact, the number of their 2-parts can be quite large (with the exception of $\textrm{M}_{22}$ where all non-linear $2$-Brauer character degrees have the same 2-part).

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 $\textrm{cd}_2(\textrm{PGL}_2(q))=\{1, q-1\}$ whenever $q=9$ or a Fermat prime, so for $p=2$, there are non-solvable groups satisfying the hypothesis of the previous theorem.

Theorem. Let $G$ be a non-solvable 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: