# The $\{2,3\}$-groups with a condition about $\mathbb{C}$-characters

Let $$G$$ be a $$\{2,3\}$$-group and $$\lvert G\rvert=2^\alpha\cdot3^\beta$$. For $$p\in\{2,3\}$$, define $$\nu_p(G)\mathrel{:=}\min\left\{\log_p\left(\frac{\lvert G\rvert}{\chi(1)}\right)_p \mathrel{\bigg\vert} \chi\in\operatorname{Irr}(G)\right\},$$ where $$\operatorname{Irr}(G)$$ is the set of all irreducible $$\mathbb{C}$$-characters of $$G$$.

Suppose that $$\nu_2(G)=1$$, $$\nu_3(G)=0$$. We want to study this group $$G$$. Especially, we want to know the answer to:

QUESTION: Are there two numbers $$M$$ and $$N$$ such that $$\alpha and $$\beta for all such groups $$G$$?

• For your question literally as stated one can take $M = \alpha + 1$ and $N = \beta + 1$, so I assume you meant "for all such groups $G$". I edited accordingly. – LSpice May 14 at 20:19

If I understand your conditions correctly, you are assuming that there is a $$3$$-block of $$G$$ of defect zero (this corresponds to an irreducible character $$\chi$$ with $$\chi(1)_{3} = |G|_{3}$$) and a $$2$$-block of $$G$$ of defect $$1$$ (for in general if a finite group $$G$$ has order divisibly by $$p^{a}$$ (but by no higher power of the prime $$p$$) and $$G$$ has an irreducible character $$\chi$$ of degree exactly divisible by $$p^{a-1}$$ (but by no higher power of $$p$$), then it is a result of R. Brauer that $$\chi$$ lies in a $$p$$-block of defect $$1$$.
Now since $$|G| = 2^{\alpha}3^{\beta}$$, we know that $$G$$ is solvable by Burnside's $$p^{a}q^{b}$$-theorem. Hence $$C_{G}(F(G)) \leq F(G)$$, where $$F(G)$$ is the unique largest nilpotent normal subgroup of $$G$$.
But for any prime divisor $$p$$ of $$|G|$$, $$O_{p}(G)$$ is contained in each defect group of each $$p$$-block of $$G$$, where $$O_{p}(G)$$ is the largest normal $$p$$-subgroup of $$G$$.
Hence we have $$O_{3}(G) = 1$$ and $$|O_{2}(G)| \leq 2$$. Since $$G$$ is a $$\{2,3\}$$-group, this implies that $$F(G)$$ has order dividing $$2$$, so that $$F(G) \leq Z(G).$$ Since $$C_{G}(F(G)) \leq F(G)$$, this forces $$G = F(G)$$ and $$|G| \in \{1,2\}$$. So the answer to your question is a resounding "yes".
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