= Solution
Conversely, the same induction-norm formula shows that $I_G(\varphi)=N$ for every nonprincipal $\varphi\in\operatorname{Irr}(N)$. Conjugation by each $g\notin N$ fixes only $1_N$, so <Brauer's permutation lemma> says that it fixes only the identity <conjugacy class> of $N$. A nonidentity element centralizing $g$ would provide another fixed class. Therefore $C_N(g)=1$ for every $g\notin N$.
We first prove that $|N|$ and $[G:N]$ are coprime. If a prime $p$ divided both, a <Sylow p-subgroup> $S$ of $G$ would have $1<S\cap N<S$. The nontrivial <normal subgroup> $S\cap N$ meets $Z(S)$ nontrivially: the class equation for the conjugation action of the <p-group> $S$ on $S\cap N$ makes the number of fixed elements divisible by $p$, and one of them is the identity. Choose $1\ne z\in N\cap Z(S)$ and $s\in S\setminus N$. Then $z\in C_N(s)$, contradicting the previous conclusion.
Thus $N$ is a <normal subgroup> that is also a <Hall subgroup>. The <Schur-Zassenhaus theorem> supplies a <group complement> $H$, meaning $G=NH$ and $N\cap H=1$. The theorem's existence assertion requires only normality and coprimality of order and index. Here $H$ is nontrivial and proper by the assumptions on $N$.
For $n\in N\setminus\{1\}$, suppose $1\ne h\in H\cap nHn^{-1}$. Write $h=nh'n^{-1}$ with $h'\in H$. Projection to $G/N$ gives $hN=h'N$, hence $h=h'$ because $H\to G/N$ is injective. This makes $n$ centralize $h\notin N$, which is impossible. Hence $H\cap nHn^{-1}=1$. Writing an arbitrary element outside $H$ as $nh_0$ reduces its conjugate intersection to this case. It follows that $H$ is a <Frobenius complement> and $G$ is a <Frobenius group>.
Moreover, for each $1\ne h\in H$, the map $v\mapsto vhv^{-1}$ from $N$ into the coset $Nh$ is injective, since $C_N(h)=1$, and hence bijective. Thus every element outside $N$ belongs to a conjugate of $H$, while no nonidentity element of $N$ does. \b[The <Frobenius kernel> is exactly $N$.] This completes both directions of the <Frobenius kernel criterion by irreducible induction>.
Back to article page
