Virtual characters are integrally generated by monomial characters from elementary subgroups

If $G=1$, then $R(G)=\mathbb{Z}\cdot 1_G$ and $1_G={\mathrm{Ind}}_G^G{1}_G$: by the definition of the trivial group as an elementary group, $G$ itself is $p$-elementary for every prime $p$. Hence every virtual character $\vartheta$ of the trivial group is an integral multiple of a monomial character induced from an elementary subgroup.

Suppose $\mathrm{\mid }G\mathrm{\mid }>1$ and write $\mathrm{\mid }G\mathrm{\mid }=p^{n_p}{l}_p$ with $p\nmid l_p$ for each prime $p\mid \mathrm{\mid }G\mathrm{\mid }$. By [F3] each $l_p{1}_G$ lies in $I_{{\mathcal{E}}_p}(G)$, hence in $I_{\mathcal{E}}(G)$ because ${\mathcal{E}}_p\subseteq \mathcal{E}$. The integers $l_p$, $p\mid \mathrm{\mid }G\mathrm{\mid }$, have greatest common divisor $1$: no prime $q\mid \mathrm{\mid }G\mathrm{\mid }$ divides $l_q$. Bézout's identity therefore provides integers $a_p$ with ${\sum }_{p\mid \mathrm{\mid }G\mathrm{\mid }}{a}_p{l}_p=1$, so $1_G={\sum }_{p\mid \mathrm{\mid }G\mathrm{\mid }}{a}_p(l_p{1}_G)\in I_{\mathcal{E}}(G)$ as this is an additive subgroup.

In either case $1_G\in I_{\mathcal{E}}(G)$ (step 1.1 for $G=1$ and step 1.2 otherwise), and by [F2] the subgroup $I_{\mathcal{E}}(G)$ is an ideal of $R(G)$. Multiplying the virtual character $\vartheta$ by $1_G$ therefore gives $\vartheta =\vartheta \cdot 1_G\in I_{\mathcal{E}}(G)$.

By [F1] there are finitely many subgroups $H_1,\dots ,H_m\in \mathcal{E}$ and virtual characters ${\theta }_i\in R(H_i)$ with $\vartheta ={\sum }_{i=1}^m{\mathrm{Ind}}_{H_i}^G{\theta }_i$. Fix such an expression and write each ${\theta }_i={\sum }_j{n}_{ij}{\theta }_{ij}$ as an integral combination of the irreducible complex characters ${\theta }_{ij}$ of $H_i$.

Each $H_i$ is $p_i$-elementary for some prime $p_i$, hence supersolvable by [F4]; [F6] therefore expresses each irreducible constituent ${\theta }_{ij}$ as ${\theta }_{ij}={\mathrm{Ind}}_{K_{ij}}^{H_i}{\lambda }_{ij}$ for a subgroup $K_{ij}\le H_i$ and a linear character ${\lambda }_{ij}$ of $K_{ij}$.

Since $K_{ij}\le H_i$ and $H_i$ is $p_i$-elementary, [F5] makes $K_{ij}$ a $p_i$-elementary, hence elementary, subgroup of $G$. So each of the subgroups $K_{ij}$ belongs to the family $\mathcal{E}$.

Consequently ${\mathrm{Ind}}_{H_i}^G{\mathrm{Ind}}_{K_{ij}}^{H_i}{\lambda }_{ij}={\mathrm{Ind}}_{K_{ij}}^G{\lambda }_{ij}$ as characters, by [F7] and the definition of the induced character. Substituting the expressions of step 4.1 into the expression of step 3.1 and collecting the integer multiplicities $n_{ij}$ expresses $\vartheta$ as an integral linear combination of the monomial characters ${\mathrm{Ind}}_{K_{ij}}^G{\lambda }_{ij}$ induced from the elementary subgroups $K_{ij}$, which is the assertion. ∎