2.1. Case that $N$ is a $p$-subgroup

We first suppose $N$ is a $p$-subgroup for some prime $p$ and induct on the order of $G$. As $H$ is supersoluble, we may find a normal subgroup $H_0 \vartriangleleft H$ of index $q$ [Reference Isaacs4, exer. 3B.9], where $q$ is the smallest prime divisor of $|{H}|$. Under the inductive hypothesis, $H_0$ is conjugate to some subgroup of $H'$ and so the left action of $H_0$ on the cosets $\varOmega =G/H'$ has a fixed point, say $n'H'$. If $H$ itself does not fix any point of $\varOmega$, then the non-empty set $\operatorname {Fix}_{\varOmega }(H_0)$ of points in $\varOmega$ fixed by $H_0$ may be partitioned into orbits of $H$, each having cardinality $[H:H_0] = q$. On the other hand, the map from $N_N(H_0)$ to $\operatorname {Fix}_{\varOmega }(H_0)$ given by $n\mapsto n\cdot n'H'$ provides a bijective correspondence between these two sets, and as $N_N(H_0) \leq N$ is a $p$-group, we may assume without loss that $q=p$. In particular, $H_0$ contains a $p'$-Hall subgroup of $H$, say $M$, such that $C_N(M) \supseteq C_N(H_0) = N_N(H_0)$ is nontrivial. Furthermore, we may assume without loss that $M\vartriangleleft H$ [Reference Isaacs4, exer. 3B.8] so $N_0 := [N,\, M] \vartriangleleft G$ where $N=N_0 \times C_N(M)$ by a theorem of Fitting [Reference Isaacs4, thm. 4.34]. Consequently, $N_0$ is a strict subgroup of $N$.

Let $P$ be a Sylow $p$-subgroup of $H$. By hypothesis, there exists some $n \in N$ such that $P^n \leq H'$. Replacing $H$ by $H^n$ in the statement of the theorem if necessary, we may assume without loss that $n$ is trivial, i.e. that $P \leq H'$.

Glauberman's lemma implies the left action of $NM/N_0$ on the cosets $G/N_0H'$ has an $M$-invariant element. As $N=N_0 \times C_N(M)$, it follows that $\overline {N} = \overline {C_N(M)}$ in $\overline G = G/N_0$. Furthermore, as fixed points come from fixed points in coprime actions [Reference Isaacs4, cor. 3.28], $\overline {N} = C_{\overline N}(M)$ so that $M$ fixes every element of $G/N_0 H'$. In particular, $M\leq N_0H'$ so that $N_0M$ acts on the cosets $N_0H'/H'$. Glauberman's lemma implies this second action also has an $M$-invariant element so that $M^{n_0} \leq H'$ for some $n_0 \in N_0$. Thus, as $H\leq N_0H'$, we may apply the inductive hypothesis in $N_0H'$ to conclude that $H$ and $H'$ are conjugate.

2.2. General case that $N$ is abelian

We again induct on the order of $G$. Building upon the results of our first case, we now assume $|{N}|$ has multiple prime divisors. In particular, we may write $N=N_1\times N_2$ where $N_1$ and $N_2$ are characteristic in $N$ and thus normal in $G$. (We may, for example, let $N_1$ be the first primary component of $N$ and $N_2$ be the product of the other primary components.) In $G/N_1$, the inductive hypothesis implies the action of $H$ on the left cosets $G/N_1H'$ has a fixed point, say $n_2N_1H'$ for some $n_2\in N_2$. Analogously, in $G/N_2$, we find that the action of $H$ on $G/N_2H'$ must also have a fixed point, say $n_1N_2H'$ for some $n_1\in N_1$. Consequently, we conclude that the action of $H$ on $G/H'$ fixes $n_1n_2H'$ so that $H$ and $H'$ are conjugate.

3.1. Proof of Corollary 1

Given a group $G$ satisfying the hypotheses of Corollary 1, we suppose by way of contradiction that $H$ and $H'$ complement $N$ in $G$ but fail to be conjugate. By Theorem 1, there exists a prime $p$ and Sylow $p$-subgroups $P$ and $P'$ of $H$ and $H'$, respectively, such that $P$ and $P'$ fail to be conjugate in $G$. Let $L$ be the unique Sylow $p$-subgroup of $N$. Then $LP$ and $LP'$ are Sylow $p$-subgroups of $G$ and thus are conjugate to the Sylow $p$-subgroup $S$ of $G$ specified in the statement of the corollary. In particular, $LP^{g_1}=L(P')^{g_2}=S$ for some $g_1,\,g_2\in G$ where $S\cap N = L$, so that by assumption, $P^{g_1}$ and $({P'})^{g_2}$ are conjugate in $G$, a contradiction.

3.2. Proof of Corollary 2

Suppose $G=N \rtimes H$ acts on a set $\varOmega$ according to the hypotheses of Corollary 2. Fix some $\alpha \in \varOmega$ and consider the point stabilizer $G_\alpha$ of $\alpha$ in $G$. As $N$ is transitive on $\varOmega$, for any $g\in G$, we have $g\cdot \alpha = n\cdot \alpha$ for some $n\in N$, so that $n^{-1}g \in G_\alpha$. Thus, $G=NG_\alpha$. As $N\vartriangleleft G$, it follows that $G_\alpha \cap N \vartriangleleft G_\alpha$.

We claim $G_\alpha$ splits over $G_\alpha \cap N$. By Gaschütz's theorem, it suffices to show that for each prime $p$, there exists a Sylow p-subgroup S of $G_\alpha$ that splits over $S\cap N$. Fix $p$, an arbitrary prime. By hypothesis, there exists some $n\in N$ and a Sylow $p$-subgroup $P$ of $H$ such that $P^n \leq G_\alpha$. Let $L$ be the Sylow $p$-subgroup of $G_\alpha \cap N$. As ${|{G_\alpha }| = |{G_\alpha \cap N}| [G:N]}$, it follows that $S=LP^n$ is a Sylow subgroup of $G_\alpha$ and $P^n$ is a complement of $S \cap N = L$ in that subgroup.

Thus, $G_\alpha$ splits and we may let $H'$ complement $G_\alpha \cap N$ in $G_\alpha$. It follows that $|{H'}| = [G_\alpha : G_\alpha \cap N] = [G:N]$ so $H'$ complements $N$ in $G$. Theorem 1 then implies that ${H'=H^n}$ for some $n\in N$ so that $H$ fixes $n \cdot \alpha$.

4.1. Conjugacy of complements

D. G. Higman's work included another result on the conjugacy of complements that we mention here for completeness. In a split extension $G\cong N\rtimes H$ of an abelian subgroup $N$, Higman considered an intermediate subgroup $N\leq B \leq G$ of index $[G:B]=b$ such that the map $n\mapsto n^b$ for $n\in N$ has an inverse. (Such an inverse always exists if $b$ and $|{N}|$ are coprime, for example.) He showed that if any two complements of $N$ in $B$ are conjugate in $B$, then any two complements of $N$ in $G$ are conjugate in $G$. Furthermore, given two specific complements $H$ and $H'$ of $N$ in $G$, he showed that $H$ and $H'$ are conjugate in $G$ if and only if $H\cap B$ and $H'\cap B$ are conjugate in $B$ [Reference Higman2, cor. 1].

4.2. Fixed points

Glauberman's original paper [Reference Glauberman1] provided two conditions for $N$-transitive actions of split extensions $G\cong N\rtimes H$ that jointly imply the existence of an $H$-invariant element, namely (S) if $W\leq G$ satisfies $WN=G$ then $W$ splits over $W\cap N$, and (Z) any two complements of $N$ in $G$ are conjugate.Footnote ² The hypotheses of our Corollary 2 do not necessarily imply (Z) but rather only that two specific complements of $N$ are conjugate, namely $H$ and a complement of $G_\alpha$ in $G_\alpha \cap N$, where $\alpha \in \varOmega$ is a member of the set on which $G$ acts. For this reason, we did not directly apply Glauberman's result but instead mirrored his proof.

To further illustrate the connection between the conjugacy of complements and the fixed points of certain group actions, we note that Corollary 2 implies Theorem 1. To see this equivalence, we may apply Corollary 2 to the action of $N\rtimes H$ on the set $\varOmega =G/H'$ of left cosets for any second complement $H'$ satisfying the hypotheses of Theorem 1.

Finally, we note that in the language of group cohomology, Corollary 1 provides a vanishing condition for the first cohomology group $H^1(H,\,N)$ of the split extension $G\cong N\rtimes H$ when $N$ is abelian and $H$ is supersoluble.