# Monomial characters of direct product

Let $$G$$ be a finite group and let $$\text{Irr}(G)$$ be the set of irreducible complex characters of $$G$$. A character $$\chi\in\text{Irr}(G)$$ is monomial if there exists a subgroup $$H\leq G$$ and a linear character $$\lambda\in\text{Lin}(H)$$ such that $$\chi=\lambda^G$$. Let $$\text{Irr}_m(G)$$ be the set of irreducible monomial characters of $$G$$. Then a group $$G$$ is monomial if $$\text{Irr}(G)=\text{Irr}_m(G)$$. I'm interested in solvable groups even if probably this is not necessary.

My question is: given two finite solvable groups $$G$$ and $$H$$ is it true that $$\text{Irr}_m(G\times H)=\{\varphi\times \psi\mid \varphi\in\text{Irr}_m(G),\psi\in\text{Irr}_m(H)\}$$

Using GAP I checked some (really small) cases and this always works.

What is this useful for? I'm interested in this question because a positive answer will give an easy way to construct a family of solvable groups $$(G_n)_{n\in\mathbb{N}}$$ such that

$$\lim\limits_{n\to \infty} \frac{|\text{Irr}_m(G_n)|}{|\text{Irr}(G_n)|}=0$$ To construct such a family fix a solvable nonmonomial group $$G$$ (e.g. $$\text{SL}_2(3)$$) and define $$G_n$$ to be the direct product of $$n$$-copies of $$G$$.

Update: The above question has a negative answer, in general, by work of van der Waall "Direct products and monomial characters". However, a positive answer is given when at least one of the factors, say $$G$$, satisfies the following property: either $$G=1$$ or every maximal subgroup $$M$$ of $$G$$ has a normal subgroup $$N\unlhd M$$ with only abelian Sylow subgroups (for every prime) and such that $$M/N$$ is nilpotent. In particular, the above family can be constructed considering $$G=\text{SL}_2(3)$$.

• Out of curiosity: How large orders did you check? It seems that comparing the numbers of monomial characters might be the quickest, but I am not sure how quick TestMonomial is. Jun 1, 2018 at 18:52
• @TobiasKildetoft: With GAP I checked the cases $G=SL_2(3)$, $H\in \{S_3,D_8,Q_8,D_{10}, A_4, D_{12}, \text{SL}_2(3), \text{GL}_2(3), \text{SmallGroup}(48,28), \text{SmallGroup}(48,32), \text{SmallGroup}(48,33), \text{SmallGroup}(72,3), \text{SmallGroup}(72,25)\}$, where $\text{SL}_2(3), \text{GL}_2(3), \text{SmallGroup}(48,28), \text{SmallGroup}(48,32), \text{SmallGroup}(48,33), \text{SmallGroup}(72,3), \text{SmallGroup}(72,25)$ are the smallest solvable nonmonomial groups. To find the monomial characters of $G$ I do Filtered(Irr(G), x-IsMonomial(x)). This method is not so fast however. Jun 2, 2018 at 15:59

Actually, $$G$$ and $$H$$ need not be solvable.