In literature [1], Isaacs introduced the concept of $B_{\pi}$-characters of a $\pi$-separable group $G$, where $\pi$ denotes a set of primes. $B_{\pi}$-characters of $G$ are also called $\pi$-Brauer characters. Denote by $B_{\pi}(G)$ the set of all $\pi$-Brauer characters of $G$. Let $\chi$ be a character of $G$ and denote by $\chi^{*}$ the restriction of $\chi$ to the set of all $\pi$-elements of $G$. In [1], Isaacs proved that the set of $\chi^{*}$ forms a basis of the space of $\pi$-class functions of $G$. In addition, he also proved that the number of $\pi$-characters of $G$ equals the number of conjugacy classes of $\pi$-elements of $G$.

Motivated by these results due to Isaacs, we want to know if other properties of Brauer characters can be generalized. In this note, by using the decomposition matrices in the $\pi$-theory of characters, the concept of principal indecomposable $B_{\pi'}$-characters of $G$ is introduced, where $\pi'$ denotes the complement set of $\pi$. Denote by $CF(G_{\pi'})$ the space of $\pi$-regular class functions of $G$. One of our main results is as follows.

Theorem A Let $G$ be a $\pi$-separable group and let $\eta_{i}\, \, (1\leq i\leq l)$ be the all the principal indecomposable $B_{\pi'}$-characters of $G$. Then the set $\{\eta_{i}|1\leq i\leq l\}$ forms a basis of $CF(G_{\pi'})$.

In the theory of Brauer character, generalized Brauer characters is introduced. It is known that the set of principal indecomposable characters forms a $\mathbb{Z}$-basis of $X(G|G_{p'})$, where $X(G)$ is the ring of all generalized characters of $G$. Let $\widetilde{\varphi}\in X(G|G_{p'})$, then $\widetilde{\varphi}(x)=0$ if $x\in G-G_{p'}$. $X(G|G_{p'})$ is a $\mathbb{Z}$-module. Similarly, we can introduce the notation $X(G|G_{\pi'})$ and the concept of generalized $B_{\pi'}$-characters of $G$. The another main result is as follows.

Theorem B Let $G$ be a $\pi$-separable group and let $\eta_{i}\, \, (1\leq i\leq l)$ be the all the principal indecomposable $B_{\pi'}$-characters of $G$. Then the set $\{\eta_{i}|1\leq i\leq l\}$ forms a $\mathbb{Z}$-basis of $X(G|G_{\pi'})$.

Throughout this paper, groups considered are finite $\pi$-separable. $Irr(G)$ denotes the set of all irreducible ordinary characters of $G$; $G_{\pi'}$ denotes the set of all $\pi$-regular elements of $G$; $\chi^{*}$ denotes the restriction of $\chi$ to $G_{\pi'}$. Let $H$ be a subgroup of $G$. Let $\chi$ and $\varphi$ be characters of $G$ and $H$ respectively. Then we write $\chi_{H}$ the restriction of $\chi$ to $H$ and $\varphi^{G}$ the lift of $\varphi$ to $G$. Unless stated otherwise, other notation and terminologies used are standard, refer to the literature [1, 4] or [9].

First we recall some basis concepts and present some lemmas which will be used in the sequel.

Definition 2.1 Let $\chi\in Irr(G)$. Then $\chi^{*}$ is said to be irreducible if $\chi^{*}$ can not be expressed as $\chi^{*}=\mu^{*}+\nu^{*}$, where $\mu, \nu$ are two class functions of $G$. $I^{\pi}(G)$ denotes the set of all irreducible $\chi^{*}$.

Lemma 2.2 [1, Corollary 9.1] Let $G$ be a $\pi$-separable group. Then $\{\chi^{*}|\chi\in B_{\pi'}(G)\}$ are linearly independent.

Lemma 2.3 [1, Corollary 9.2] Let $G$ be a finite $\pi$-separable group. Then $|B_{\pi'}(G)|$ equals the number of $\pi$-regular classes of $G$.

Lemma 2.4 [1, Corollary 10.2] Let $G$ be a finite $\pi$-separable group. Then the map $\phi: B_{\pi'}(G)\rightarrow I^{\pi}(G)$ defined by $\Psi\mapsto \Psi^{*}$ is a bijection. In particular, $I^{\pi}(G)$ is a basis of $CF(G_{\pi'})$.

Lemma 2.5 [1, Corollary 10.3] Let $G$ be a $p$-solvable. Then $\phi: B_{p'}(G)\rightarrow IBr(G)$ defined by $\Psi\mapsto \Psi^{*}$ is a bijection, where $IBr(G)$ denotes the set of all irreducible Brauer characters of $G$.

Remark 2.6 For simplicity of notation, in view of Lemmas 2.4 and 2.5, we may write $B_{\pi'}(G)=I^{\pi}(G)$. In particular, $B_{p'}(G)=IBr(G)$ when $\pi=\{p\}$. Thus the space of all $\pi$-regular class functions are a generalization of the space of Brauer $p$-regular class functions of $G$.

Lemma 2.7 [1, Corollary 10.1] Let $G$ be a finite $\pi$-separable group, $\xi\in Irr(G)$ and $\eta\in B_{\pi'}(G)$. Then there exists a nonnegative integral "decomposition number" $d_{\xi\eta}$ such that $\xi^{*}=\sum\limits_{\eta\in B_{\pi'}(G)}d_{\xi\eta}\eta$ for any $\xi\in Irr(G)$.

Set $Irr(G)=\{\chi_{1}, \chi_{2}, \cdots, \chi_{k}\}$, $B_{\pi'}(G)=\{\varphi_{1}, \varphi_{2}, \cdots, \varphi_{l}\}$. By Lemma 2.7,

$ (\chi_{1}^{*}, \chi_{2}^{*}, \cdots, \chi_{k}^{*})^{t}=(d_{ij})_{k\times l}(\varphi_{1}, \varphi_{2}, \cdots, \varphi_{l})^{t}. $

Write $D=(d_{ij})_{k\times l}$, $C=D^{t}D$. Then $D$ and $C$ are said to be the decomposition matrix and the Cartan matrix of $G$, respectively.

Let $(\eta_{1}, \eta_{2}, \cdots, \eta_{l})^{t}=^{t}D(\chi_{1}, \chi_{2}, \cdots, \chi_{k})^{t}, $ then $\eta_{j}\, \, (1\leq j \leq l)$ are said to be the principal indecomposable $B_{\pi'}$-characters of $G$. It is easy to verify that the relation of the principal indecomposable $B_{\pi'}$-characters and the $B_{\pi'}$-characters of $G$ is as follows

$ (\eta_{1}^{*}, \eta_{2}^{*}, \cdots, \eta_{l}^{*})^{t}=C(\varphi_{1}, \varphi_{2}, \cdots, \varphi_{l})^{t}. $

In particular, if $G$ is a $\pi'$-group, then $\chi_{i}=\eta_{i}=\varphi_{i}$, where $1\leq i\leq k$.

Lemma 2.8 Let $G$ be a finite $\pi$-separable group and notation be as above, then the following statements hold

(1) $\sum\limits_{i=1}^{l}\overline{\eta_{i}(x)}\varphi_{i}(y)=\delta_{x^{G}y^{G}}|C_{G}(x)|$ for any $y\in G_{\pi'}$;

(2) $\eta_{i}(x)=0$ for any $x\in G-G_{\pi'}$, where $1\leq i\leq l$.

Proof (1) For any $y\in G_{\pi'}$, we have $\chi_{i}(y)=\sum\limits_{j=1}^{l}d_{ij}\varphi_{j}(y)$. By the second orthogonal relation of characters, one gets that

$ \begin{eqnarray*}\delta_{x^{G}y^{G}}|C_{G}(x)|&=&\sum\limits_{i=1}^{k}\overline{\chi_{i}(x)}\chi_{i}(y) =\sum\limits_{i=1}^{k}\overline{\chi_{i}(x)}\sum\limits_{j=1}^{l}d_{ij}\varphi_{j}(y) ={\sum\limits_{i=1}^{k}\sum\limits_{j=1}^{l}d_{ij}\overline{\chi_{i}(x)}\varphi_{j}(y)}\\ &=&{\sum\limits_{j=1}^{k}d_{ji}}\sum\limits_{i=1}^{l}\overline{\chi_{j}(x)}\varphi_{i}(y) ={\sum\limits_{i=1}^{l}d_{ji}}(\sum\limits_{j=1}^{k}\overline{\chi_{j}(x)})\varphi_{i}(y) =\sum\limits_{i=1}^{l}\overline{\eta_{i}(x)}\varphi_{i}(y).\end{eqnarray*} $

Consequently, we have $\sum\limits_{i=1}^{l}\overline{\eta_{i}(x)}\varphi_{i}(y)=\delta_{x^{G}y^{G}}|C_{G}(x)|$.

(2) For any $x\in G-G_{\pi'}$ and $y\in G_{\pi'}$, by (1), we have $\sum\limits_{i=1}^{l}\overline{\eta_{i}(x)}\varphi_{i}(y)=0$. As $y$ is arbitrary, we obtain that $\sum\limits_{i=1}^{l}\overline{\eta_{i}(x)}\varphi_{i}=0$. Note that $\{\varphi_{i}|1\leq i\leq l\}$ is linearly independent, so the previous equality yields that $\eta_{i}(x)=0$ for each $1\leq i\leq l$. This completes the proof of Lemma 2.8.

Let $G$ be a $\pi$-separable group and let $\varphi$ and $\eta$ be class functions defined on $G_{\pi'}$. Write $(\varphi, \eta)'=\frac{1}{|G|}\sum\limits_{x\in G_{\pi'}}\varphi(x)\overline{\eta(x)}$.

Lemma 2.9 Let $G$ be a $\pi$-separable group and notation be as above, then $(\eta_{i}, \varphi_{j})'=\delta_{ij}.$

Proof By Lemma 2.3 we may assume that $Cl(G_{\pi'})=\{C_{1}, C_{2}, \cdots, C_{l}\}$, where $x_{i}\in C_{i}$ $(1\leq i \leq l)$.

Write $\Phi=(\varphi_{i}(x_{j}))_{l\times l}$, $Y=(\eta_{i}(x_{j}))_{l\times l}$ and $S=\left( \begin{array}{ccc} |C_{G}(x_{1})|&\cdots&0 \\ \vdots&\ddots&\vdots \\ 0&\ldots&|C_{G}(x_{l})| \\ \end{array} \right)$.

By Lemma 2.8(1), we have $\overline{Y}^{t}\Phi =S$ and thus $\overline{Y}^{t}(\Phi S^{-1})=I$, i.e., $(\Phi S^{-1})^{t}\overline{Y}=I$. Therefore $\sum\limits_{\nu=1}^{l}\varphi_{i}(x_{\nu})\frac{1}{|C_{G}(x_{\nu})|}\overline{\eta_{j}(x_{\nu})}=\delta_{ij}$, i.e.,

$ \frac{1}{|G|}\sum\limits_{\nu=1}^{l}\varphi_{i}(x_{\nu})\frac{|G|}{|C_{G}(x_{\nu})|}\overline{\eta_{j}(x_{\nu})}=\delta_{ij}. $

On the other hand,

$ \begin{eqnarray*}\frac{1}{|G|}\sum\limits_{\nu=1}^{l}\varphi_{i}(x_{\nu})\frac{|G|}{|C_{G}(x_{\nu})|}\overline{\eta_{j}(x_{\nu})} &=&\frac{1}{|G|}\sum\limits_{\nu=1}^{l}|Cl(x_{\nu})|\varphi_{i}(x_{\nu})\overline{\eta_{j}(x_{\nu})}\\ &=&\frac{1}{|G|}\sum\limits_{x\in G_{\pi'}}\varphi_{i}(x)\overline{\eta_{j}(x)}=(\varphi_{i}, \eta_{j})'.\end{eqnarray*} $

Consequently, $(\varphi_{i}, \eta_{j})'=\delta_{ij}.$

Lemma 2.10 Let $G$ be a $\pi$-separable group and notation be as above, then the following statements hold

(1) $C_{ij}=(\eta_{i}, \eta_{j})_{G}$;

(2) Let $Z=((\varphi_{i}, \varphi_{j})')_{l\times l}$, then $CZ=I$.

Proof (1) Since $C=^{t}DD$, it follows that $C_{ij}=\sum\limits_{v=1}^{k}d_{vi}d_{vj}$. But note that $\eta_{i}=\sum\limits_{j=1}^{k}d_{ji}\chi_{j}\, \, (1\leq i \leq l), $ so

$ \begin{eqnarray*}(\eta_{i}, \eta_{j})_{G}&=&\frac{1}{|G|}\sum\limits_{x\in G}\eta_{i}(x)\overline{\eta_{j}(x)} =\frac{1}{|G|}\sum\limits_{x\in G}(\sum\limits_{u=1}^{k}d_{ui}\chi_{u}(x)\sum\limits_{v=1}^{k}d_{vj}\overline{\chi_{v}(x)})\\ &=&{\sum\limits_{u=1}^{k}}\sum\limits_{v=1}^{k}d_{ui}d_{vj}(\frac{1}{|G|}\sum\limits_{x\in G}\chi_{u}(x)\overline{\chi_{v}(x)})={\sum\limits_{u=1}^{k}}\sum\limits_{v=1}^{k}d_{ui}d_{vj}(\chi_{u}, \chi_{v})_{G}\\ &=&{\sum\limits_{u=1}^{k}}\sum\limits_{v=1}^{k}d_{ui}d_{vj}\delta_{uv}=\sum\limits_{u=1}^{k}d_{ui}d_{vj}=C_{ij}.\end{eqnarray*} $

(2) Since $CZ=\left( \begin{array}{ccc} C_{11} & \cdots & C_{1l} \\ \vdots & \ddots & \vdots \\ C_{l1} & \cdots & C_{ll} \\ \end{array} \right) \left( \begin{array}{ccc} (\varphi_{1}, \varphi_{1})' &\cdots &(\varphi_{1}, \varphi_{1})' \\ \vdots& \ddots& \vdots \\ (\varphi_{l}, \varphi_{1})'&\cdots&(\varphi_{l}, \varphi_{l})' \\ \end{array} \right)$, it follows that the $(i, j)$-element of $CZ$ is $\sum\limits_{\nu=1}^{l}C_{i\nu}(\varphi_{v}, \varphi_{j})'$. It suffices to show that $\sum\limits_{\nu=1}^{l}C_{i\nu}(\varphi_{v}, \varphi_{j})'=\delta_{ij}$. In fact, we have

$ \begin{eqnarray*}&&\sum\limits_{\nu=1}^{l}C_{i\nu}(\varphi_{v}, \varphi_{j})'=\sum\limits_{\nu=1}^{l} (\eta_{i}, \eta_{\nu})_{G}(\varphi_{v}, \varphi_{j})'\\ &=&\sum\limits_{\nu=1}^{l}\frac{1}{|G|}\sum\limits_{x\in G}\eta_{i}(x)\overline{\eta_{\nu}(x)}\frac{1}{|G|} \sum\limits_{y\in G_{\pi'}}\varphi_{\nu}(y)\overline{\varphi_{j}(y)}=\frac{1}{|G|^{2}}\sum\limits_{\nu=1}^{l}\sum\limits_{x\in G_{\pi'}}\sum\limits_{y\in G_{\pi'}} \eta_{i}(x)\overline{\eta_{\nu}(x)}\varphi_{\nu}(y)\overline{\varphi_{j}(y)}\\ &=&\frac{1}{|G|^{2}}\sum\limits_{x\in G_{\pi'}}\sum\limits_{y\in G_{\pi'}}(\sum\limits_{\nu=1}^{l}\overline{\eta_{\nu}(x)}\varphi_{\nu}(y)) \eta_{i}(x)\overline{\varphi_{j}(y)}=\frac{1}{|G|^{2}}\sum\limits_{x\in G_{\pi'}}\sum\limits_{y\in G_{\pi'}}(\delta_{x^{G}, y^{G}}|C_{G}(x)|) \eta_{i}(x)\overline{\varphi_{j}(y)}\\ &=&\frac{1}{|G|^{2}}\sum\limits_{x\in G_{\pi'}}\frac{|G|}{|C_{G}(x)|}|C_{G}(x)| \eta_{i}(x)\overline{\varphi_{j}(x)}=\frac{1}{|G|}\sum\limits_{x\in G_{\pi'}} \eta_{i}(x)\overline{\varphi_{j}(x)}=(\eta_{i}, \varphi_{j})'=\delta_{ij}.\end{eqnarray*} $

Lemma 2.11 [9, Theorem 4.2] Let $\varphi\in CF(G)$. Then $\varphi\in X(G)$ if and only if $\varphi_{E}\in X(E)$ for any $E\in \varepsilon (G)$, where $\varepsilon (G)$ denotes the set of elementary subgroups of $G$.

Theorem 3.1 Let $G$ be a $\pi$-separable group and $\{\eta_{i}|1\leq i\leq l\}$ be principal indecomposable $B_{\pi '}$-characters. Then $\{\eta_{i}|1\leq i\leq l\}$ is a basis of the space $CF(G_{\pi'})$ of all $\pi$-regular functions.

Proof We first show that $\{\eta_{i}|1\leq i\leq l\}$ is linearly independent. Let $\sum\limits_{i=1}^{l}k_{i}\eta_{i}=0$, where $k_{i}\in {\mathbb{C}}$. We shall show that $k_{i}=0$.

By Lemma 2.9, $(\varphi, \sum\limits_{j=1}^{l}k_{j}\eta_{j})'=\sum\limits_{j=1}^{l}k_{j}(\varphi_{i}, \eta_{j})'=\sum\limits_{j=1}^{l}k_{j}\delta_{ij}=k_{i}$, it follows that $k_{i}=0$. Therefore $\{\eta_{i}|1\leq i\leq l\}$ is linearly independent. Note that $B_{\pi'}(G)$ is a basis of $CF(G_{\pi'})$, so $|B_{\pi'}(G)|=l$ and hence $\{\eta_{i}|1\leq i\leq l\}$ is also a basis of the space $CF(G_{\pi'})$ of all $\pi$-regular functions.

Theorem 3.2 Let $G$ be a $\pi$-separable group, $\varphi\in B_{\pi'}(G)$ and $x\in G$. Define $\widetilde{\varphi}$ as follows: $\widetilde{\varphi}(x)=\varphi(x_{\pi'})$. Then $\widetilde{\varphi}$ is a generalized character of $G$. It follows that there exists $m_{i}\in {\mathbb{Z}}$ such that $\widetilde{\varphi}=\sum\limits_{i=1}^{k}m_{i}\chi_{i}$. We call $\widetilde{\varphi}$ a generalized $B_{\pi'}$-character of $G$.

Proof Let $E$ be a elementary group of $G$. Then $E$ is nilpotent and hence $E=P\times Q$, where $P$ is a $\pi$-group and $Q$ is a $\pi'$-group. By the definition of $\widetilde{\varphi}$, $\widetilde{\varphi}_{E}=1_{P}\times \varphi_{Q}$. In fact, for any $x\in E$, we have $x=pq$ with $p\in P$ and $q\in Q$. Note that $\widetilde{\varphi}_{E}(x)=\widetilde{\varphi}_{E}(pq)=\varphi(q)$ and $(1_{p}\times \varphi_{Q})(x)=(1_{p}\times \varphi_{Q})(pq)=1_{p}(p)\times\varphi_{Q}(q)=\varphi (q)$, so $\widetilde{\varphi}_{E}=1_{p}\times \varphi_{Q}$. Hence $\widetilde{\varphi}_{E}$ is a character on $E$, i.e., $\widetilde{\varphi}_{E}\in X(E)$. By Lemma 2.11, $\widetilde{\varphi}\in X(G)$. It follows that there exists $m_{i}\in \mathbb{Z}$ such that $\widetilde{\varphi}=\sum\limits_{i=1}^{l}m_{i}\chi_{i}$.

Theorem 3.3 Let $G$ be a $\pi$-separable group and notation as above. Then the rank of the decomposition matrix $D$ equals $l$.

Proof Write $\widetilde{\varphi_{i}}=\sum\limits_{j}m_{ij}\chi_{j}$ with $m_{ij}\in {\mathbb{Z}}$ and let $M=(m_{ij})_{l\times k}$. Since

$ \varphi_{i}=\widetilde{\varphi_{i}|}_{G_{\pi'}}=\sum\limits_{j=1}^{k}m_{ij}(\chi_{j}|_{G_{\pi'}}) =\sum\limits_{j=1}^{k}m_{ij}(\sum\limits_{r=1}^{l}d_{jr}\varphi_{r})=\sum\limits_{r=1}^{l}(\sum\limits_{j=1}^{k}m_{ij}d_{jr})\varphi_{r}. $

Since $\{\varphi_{i}\}$ is linearly independent, it follows that $\sum\limits_{j=1}^{k}m_{ij}d_{jr}=1$, i.e., $MD=I$. Therefore the rank of $D$ is $l$.

Theorem 3.4 Let $G$ be a $\pi$-separable group and $\{\eta_{i}|1\leq i\leq l\}$ be principal indecomposable $B_{\pi '}$-characters. Then $\{\eta_{i}|1\leq i\leq l\}$ is a $\mathbb{Z}$-basis of $X(G|G_{\pi'})$.

Proof Note that by Lemma 2.8(2), $\eta_{i}(x)=0$ for any $x\in G-G_{\pi'}$, so $\eta_{i}\in X(G|G_{\pi'})\neq \emptyset$. Since $\eta_{i}=\sum\limits_{j=1}^{k}d_{ji}\chi_{j}$ ($1\leq i\leq l$), ${\rm rank}(D)=l$ and $\{\chi_{i}|1\leq i\leq k\}$ is linearly independent, it follows that $\{\eta_{i}|1\leq i\leq l\}$ is also linearly independent. Since $C\otimes_{\mathbb{Z}} X(G|G_{\pi'})$ is a subspace of $CF((G_{\pi'}))$, so ${\rm dim}_{C}(C\otimes_{\mathbb{Z}} X(G|G_{\pi'}))\leq l$ and hence $\{\eta_{i}|1\leq i\leq l\}$ is a $C$-basis of $C\otimes_{\mathbb{Z}} X(G|G_{\pi'})$. For any $\theta\in X(G|G_{\pi'})$, we have $\theta=\sum\limits_{i=1}^{l}a_{i}\eta_{i}$ ($a_{i}\in C$) and hence

$ \begin{eqnarray*}a_{i}&=&(\theta, \varphi_{i})'=\frac{1}{|G|}\sum\limits_{x\in G_{\pi'}}\theta(x)\overline{\varphi_{i}(x)}=\frac{1}{|G|}\sum\limits_{x\in G}\theta(x)\overline{\widetilde{\varphi_{i}}(x)}\\ &=&(\theta, \widetilde{\varphi_{i}})_{G}=(\theta, \sum\limits_{j}m_{ij}\chi_{j})_{G}=\sum\limits_{j}m_{ij}(\theta, \chi_{j})_{G}.\end{eqnarray*} $

Note that $m_{ij}\in {\mathbb{Z}}$, $\theta\in X(G|G_{\pi'})$ and $(\theta, \chi_{j})_{G}\in {\mathbb{Z}}$, so $a_{i}\in {\mathbb{Z}}$. Therefore $\{\eta_{i}|1\leq i\leq l\}$ is a $\mathbb{Z}$-basis of $X(G|G_{\pi'})$.