# Fixed point conditions for non-coprime actions

Michael C Burkhart

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Michael C Burkhart. Fixed point conditions for non-coprime actions. Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, In press, 10.1017/prm.2023.96 hal-04217840

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# Fixed point conditions for non-coprime actions

Michael C. Burkhart

University of Cambridge, Cambridge, United Kingdom

(mcb93@cam.ac.uk)

(Received 13 March 2023; accepted 20 August 2023)

In the setting of finite groups, suppose $J$ acts on $N$ via automorphisms so that the induced semidirect product $N \times J$ acts on some non-empty set $\Omega$, with $N$ acting transitively. Glauberman proved that if the orders of $J$ and $N$ are coprime, then $J$ fixes a point in $\Omega$. We consider the non-coprime case and show that if $N$ is abelian and a Sylow $p$-subgroup of $J$ fixes a point in $\Omega$ for each prime $p$, then $J$ fixes a point in $\Omega$. We also show that if $N$ is nilpotent, $N \times J$ is supersoluble, and a Sylow $p$-subgroup of $J$ fixes a point in $\Omega$ for each prime $p$, then $J$ fixes a point in $\Omega$.

**Keywords:** Non-coprime actions; conjugacy of complements; supersoluble groups

2020 Mathematics Subject Classification 05E18; 20E22; 20E45; 20F16; 20J06; 55N45

## 1. Introduction

Suppose a finite group $J$ acts via automorphisms on a finite group $N$ and the induced semi-direct product $G = N \times J$ acts on some non-empty set $\Omega$ where the action of $N$ is transitive. Glauberman showed that if each supplement $H$ of $N$ in $G$ splits over $N \cap H$ and each complement of $N$ in $G$ is conjugate to $J$, then there exists a $J$-invariant element $\omega \in \Omega$. Consequently, if the orders of $J$ and $N$ are coprime so that the Schur–Zassenhaus theorem applies, a fixed point always exists [4, Thm. 4]. In this note, we consider the non-coprime case and establish some conditions for the existence of a fixed point.

Given an action as described above, consider the stabiliser $G_\alpha \le G$ fixing an arbitrary point $\alpha \in \Omega$. As $N$ is transitive, $G_\alpha$ supplements $N$ in $G$. In this context, $J$ fixes an element of $\Omega$ if and only if the following two conditions are met. Firstly, we must ensure $G_\alpha$ splits over $N \cap G_\alpha$ so that there exists some complement $J'$. As $G/N \cong G_\alpha/(N \cap G_\alpha)$, it will follow that $J'$ also complements $N$ in $G$. Secondly, we require that $J' = g^{-1}Jg$ for some $g \in G$ so that $J$ fixes $g \cdot \alpha$. For the latter requirement, we concern ourselves with conditions for two specific complements in a semidirect product to be conjugate.

To this end, we say two subgroups $H$ and $H'$ are locally conjugate in a group $G$ if for each prime $p$, a Sylow $p$-subgroup of $H$ is conjugate to a Sylow $p$-subgroup of $H'$. Losey and Stonehewer showed that if $H$ and $H'$ are locally conjugate supplements of some normal nilpotent subgroup $N$ in a soluble group $G$, then $H$ and $H'$ are

© The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
conjugate if either $G/N$ is nilpotent or $N$ is abelian [7]. Evans and Shin further showed that if $N$ is abelian, then $G$ need not be soluble [3].

We first restrict $N$ to be abelian and use a decomposition result from group cohomology to provide an alternate proof of:

**LEMMA 1.1** (Evans and Shin). In a finite group, two complements of a normal abelian subgroup are conjugate if and only if they are locally conjugate.

We use this, along with Gaschütz's result that a finite group $G$ splits over an abelian subgroup $N$ if and only if for each prime $p$, a Sylow $p$-subgroup $S$ of $G$ splits over $N \cap S$, to show:

**THEOREM 1.2.** *Given a finite group $J$ acting via automorphisms on a finite abelian group $N$, suppose the induced semidirect product $N \times J$ acts on some non-empty set $\Omega$ where the action of $N$ is transitive. If for each prime $p$, a Sylow $p$-subgroup of $J$ fixes an element of $\Omega$, then there exists some $J$-invariant element $\omega \in \Omega$.*

This had previously been shown using elementary arguments for the special case that $J$ is supersoluble [2, Cor. 2]. The theorem implies:

**COROLLARY 1.3.** Let $G$ be a finite split extension over an abelian subgroup $N$. If for each prime $p$ there is a Sylow $p$-subgroup $S$ of $G$ such that any two complements of $N \cap S$ in $S$ are conjugate, then any two complements of $N$ in $G$ are $G$-conjugate.

This extends a result of D. G. Higman [5, Cor. 2] that requires the complements of $N \cap S$ in $S$ to be conjugate within $S$.

We then consider nilpotent $N$ and supersoluble $N \times J$. We adapt our approach for lemma 1.1 to nonabelian cohomology and demonstrate:

**LEMMA 1.4.** In a finite supersoluble group, two complements of a normal nilpotent subgroup are conjugate if and only if they are locally conjugate.

With this, we then show:

**THEOREM 1.5.** *Given a finite group $J$ acting via automorphisms on a finite nilpotent group $N$, suppose the induced semidirect product $N \times J$ is supersoluble and acts on some non-empty set $\Omega$ where the action of $N$ is transitive. If for each prime $p$, a Sylow $p$-subgroup of $J$ fixes an element of $\Omega$, then there exists some $J$-invariant element $\omega \in \Omega$.*

The theorem also implies an analogue of corollary 1.3 that we state and prove in § 3.

## 1.1. Outline

We proceed as follows. In the remainder of this section, we introduce notation and some conventions from group cohomology. In the next section, we restrict $N$ to be abelian and prove theorem 1.2. We then restrict $N$ to be nilpotent and $N \times J$ to be supersoluble in § 3 and prove theorem 1.5, before concluding in § 4.
## 1.2. Notation and conventions

All groups in this note are assumed finite. A subgroup $K \le G$ supplements $N \triangleleft G$ if $G = NK$ and complements $N$ if it both supplements $N$ and the intersection $N \cap K$ is trivial. We denote conjugation by $g^\gamma = \gamma^{-1}g\gamma$ for $g, \gamma \in G$ and otherwise let groups act from the left. For a prime $p$, we let $\mathrm{Syl}_p(G)$ denote the set of Sylow $p$-subgroups of a group $G$.

We rely on rudimentary notions from group cohomology that can be found in the texts of Brown [1] and Serre [8]. Given a group $J$ acting on a group $N$ via automorphisms, crossed homomorphisms or 1-cocycles are maps $\varphi: J \to N$ satisfying $\varphi(jj') = \varphi(j)\varphi(j')^{j^{-1}}$ for all $j, j' \in J$. Two such maps $\varphi$ and $\varphi'$ are cohomologous if there exists $n \in N$ such that $\varphi'(j) = n^{-1}\varphi(j)n^{j^{-1}}$ for all $j \in J$; in this case, we write $\varphi \sim \varphi'$. We take the first cohomology $H^1(J, N)$ to be the pointed set $Z^1(J, N)$ of crossed homomorphisms modulo this equivalence. The distinguished point corresponds to the equivalence class containing the map taking each element of $J$ to the identity of $N$. Our interest in this set stems primarily from the well-known bijec- tive correspondence [8, Exer. 1 in §I.5.1] between it and the $N$-conjugacy classes of complements to $N$ in $N \times J$. Specifically, for each $\varphi \in Z^1(J, N)$, the subgroup $F(\varphi) = \{\varphi(j)j\}_{j \in J}$ complements $N$ in $NJ$ and all such complements may be writ- ten in this way. Two crossed homomorphisms yield conjugate complements under $F$ if and only if they are cohomologous, so $F$ induces the desired correspondence.

For a subgroup $K \le J$, we let $\varphi|_K$ denote the restriction of $\varphi \in Z^1(J, N)$ to $K$ and $\mathrm{res}_K^J: H^1(J, N) \to H^1(K, N)$ be the map induced in cohomology. For $\varphi \in Z^1(K, N)$ and $j \in J$, define $\varphi^j(x) = \varphi(x^{j^{-1}})^j$. We call $\varphi$ $J$-invariant if $\mathrm{res}_{K \cap K^j}^K \varphi \sim \mathrm{res}_{K \cap K^j}^{K^j} \varphi^j$ for all $j \in J$ and let $\mathrm{inv}_J H^1(K, N)$ denote the set of $J$-invariant elements in $H^1(K, N)$. For any $\varphi \in Z^1(J, N)$, we have $\varphi^j(x) = n^{-1}\varphi(x)n^{j^{-1}}$ where $n = \varphi(j^{-1})$ so that $\varphi^j \sim \varphi$. In particular, $\mathrm{res}_K^J H^1(J, N) \subseteq \mathrm{inv}_J H^1(K, N)$.

## 2. N is abelian

In this section, we restrict $N$ to be abelian so that $H^1(J, N)$ takes the form of an abelian group. We first prove lemma 1.1 as stated in § 1.

*Proof of lemma 1.1.* Suppose we are given locally conjugate complements $J$ and $J'$ of a normal abelian subgroup $N$ in some group $G$. As any element $g \in G$ may be uniquely written $g = jn$ for $j \in J$ and $n \in N$, for each prime $p$ we have $J'_p = (J_p)^n$ for some $J_p \in \mathrm{Syl}_p(J)$, $J'_p \in \mathrm{Syl}_p(J')$, and $n \in N$. Let $\varphi' \in Z^1(J, N)$ denote the crossed homomorphism corresponding to $J'$. It suffices to show that $\varphi' \sim 1$, where $1 \in Z^1(J, N)$ denotes the map taking each element of $J$ to the identity of $N$. Through the $p$-primary decomposition of $H^1(J, N)$, we have the isomorphism [1, §III.10]:

$$H^1(J, N) \cong \bigoplus_{p \in \mathcal{D}} \mathrm{inv}_J H^1(J_p, N) \qquad (2.1)$$

where $\mathcal{D}$ is the set of prime divisors of $|J|$ and the $J_p$ are those given above. For every $p \in \mathcal{D}$, we see that $\varphi'|_{J_p} \sim 1|_{J_p}$ as $J_p$ and $J'_p$ are $N$-conjugate complements of $N$ in $NJ_p$. Thus, $\varphi'$ maps to the identity in each direct summand on the right-hand side of (2.1) and we may conclude $\varphi' \sim 1$ so that $J$ and $J'$ are conjugate. $\square$
We can now use the lemma and Gaschütz's theorem to prove theorem 1.2.

*Proof of theorem 1.2.* Given $J$, $N$, and $\Omega$ as described in the hypotheses of the theorem, let $G = N \times J$ denote the induced semidirect product and consider the stabiliser subgroup $G_\alpha$ for some fixed $\alpha \in \Omega$. As $N$ acts transitively, any $g \in G$ may be written $g \cdot \alpha = n \cdot \alpha$ for some $n \in N$, so that $n^{-1}g \in G_\alpha$. Thus, $G = NG_\alpha$.

We claim $G_\alpha$ splits over $N \cap G_\alpha$. For any prime $p$, there exists by hypothesis some $n \in N$ and $P \in \text{Syl}_p(J)$ such that $P^n \le G_\alpha$. Let $L \in \text{Syl}_p(N \cap G_\alpha)$. As $|G_\alpha| = |N \cap G_\alpha| [G : N]$, it follows that $S = LP^n \in \text{Syl}_p(G_\alpha)$ so $P^n$ complements $S \cap N = L$ in $S$. As the choice of prime $p$ was arbitrary, we may apply Gaschütz's theorem to conclude that $G_\alpha$ splits over $N \cap G_\alpha$.

Let $J'$ complement $N \cap G_\alpha$ in $G_\alpha$. As $G/N \cong G_\alpha/(N \cap G_\alpha)$, it follows that $J'$ also complements $N$ in $G$. Lemma 1.1 then implies that $J' = J^g$ for some $g \in G$ so that $J$ fixes $\omega = g \cdot \alpha$. $\square$

Finally, we outline how corollary 1.3 follows from theorem 1.2.

*Proof of corollary 1.2.* Given a group $G$ satisfying the hypotheses of the corollary, suppose $J$ and $J'$ each complement $N$ in $G$. Then $G$ acts on the cosets $\Omega = G/J'$ in such a way that we may apply theorem 1.2 to infer that $J$ fixes $gJ'$ for some $g \in G$. Therefore, $J$ and $J'$ are conjugate. As the choice of complements was arbitrary, we may conclude. $\square$

### 3. $N$ is nilpotent and $N \times J$ is supersoluble

In this section, we suppose that $N$ is nilpotent and $N \times J$ is supersoluble. Consequently, $N$ decomposes as the direct sum $N \cong \bigoplus_{p \in D} N_p$ over its characteristic Sylow $p$-subgroups $N_p$ where $D$ denotes the set of prime divisors of $|N|$. Direct calculations show that the natural projections $N \to N_p$ induce an isomorphism of pointed sets

$$H^1(J, N) \cong \bigoplus_{p \in D} H^1(J, N_p). \qquad (3.1)$$

To parse the components on the right-hand side of (3.1), we introduce the following:

**PROPOSITION 3.1.** Suppose a group $J$ acts on a $p$-group $N$ via automorphisms, so that the induced semidirect product $N \times J$ is supersoluble. Then $\text{res}_{J_p}^J : H^1(J, N) \to \text{inv}_J H^1(J_p, N)$ is an isomorphism for $J_p \in \text{Syl}_p(J)$.

*Proof.* We induct on the order of $J$. If $J$ itself is a $p$-group, the conclusion is immediate. If $p$ is not a divisor of $|J|$, the lemma follows from the Schur–Zassenhaus theorem. Otherwise, let $Q \subsetneq J$ be a Sylow $q$-subgroup where $q$ is the largest prime divisor of $|J|$ [6, Exer. 3B.10] so that $J \cong Q \times M$ for some Hall $q'$-subgroup $M \le J$. Consider the inflation–restriction exact sequence [8, §I.5.8],

$$1 \to H^1(J/Q, N^Q) \to H^1(J, N) \xrightarrow{\text{res}_Q^J} H^1(Q, N)^{J/Q} \qquad (3.2)$$

where $N^Q$ denotes the elements of $N$ fixed by $Q$.
If $q \neq p$, then $H^1(Q, N)$ is trivial so that $H^1(J, N) \cong H^1(M, N^Q)$. In the supersoluble group $NQ$, $Q$ is a Sylow $q$-subgroup for the largest prime divisor of $|NQ|$, so that $Q \triangleleft NQ$ and $N^Q = N$. Consequently, $H^1(J, N) \cong H^1(M, N)$. We claim that $\text{res}_M^J$ affords this isomorphism. It suffices to show that $\text{res}_M^J$ is surjective. For any $\varphi \in Z^1(M, N)$, we may define $\tilde{\varphi}: J \to N$ by $\tilde{\varphi}(qm) = \varphi(m)$ for $q \in Q$ and $m \in M$. This map is well-defined as $J \cong Q \times M$. For $q, q' \in Q$ and $m, m' \in M$, we have $\tilde{\varphi}(qmq'm') = \varphi(mm') = \varphi(m)\varphi(m')^{m-1} = \tilde{\varphi}(qm)\tilde{\varphi}(q'm')(qm)^{-1}$, where the last equality follows from the fact that elements of $N$ commute with elements of $Q$. Thus, $\tilde{\varphi} \in Z^1(J, N)$. As $\tilde{\varphi}|_M = \varphi$, we conclude $\text{res}_M^J$ is surjective.

Exchanging $M$ for a conjugate if necessary, we may assume that $J_p \leq M$. As $\text{res}_{J_p}^M$ is injective by induction, it follows that the composition $\text{res}_{J_p}^J = \text{res}_{J_p}^M \circ \text{res}_M^J$ is also injective. On the other hand,

$$ \text{inv}_J H^1(J_p, N) \subseteq \text{inv}_M H^1(J_p, N) = \text{res}_{J_p}^M H^1(M, N) \subseteq \text{res}_{J_p}^J H^1(J, N) $$

where the equality above follows from the inductive hypothesis, so that $\text{res}_{J_p}^J$ is surjective.

Otherwise, $q = p$, so that $J_p = Q$ is a Sylow $p$-subgroup of $J$. In this case, $H^1(J/Q, N^Q)$ is trivial in (3.2) and so $\text{res}_{J_p}^J$ is injective. As $H^1(Q, N)^{J/Q} = \text{inv}_J H^1(J, N)$, it remains to show that this map is surjective. For $M$-invariant $\varphi \in Z^1(J_p, N)$, define $\tilde{\varphi}: J \to N$ by $\tilde{\varphi}(hm) = \varphi(h)$ for $h \in J_p$ and $m \in M$. Then for any $h, h' \in J_p$ and $m, m' \in M$, we have $\tilde{\varphi}(hmh'm') = \varphi(h(h')^{m-1}) = \varphi(h)\varphi((h')^{m-1})^{h-1} = \varphi(h)\varphi(h')^{m-1}h^{-1} = \tilde{\varphi}(hm)\tilde{\varphi}(h'm')^{(hm)^{-1}}$ where the third equality follows from $\varphi$ being $M$-invariant. As $J \cong J_p \times M$, we conclude that $\tilde{\varphi} \in Z^1(J, N)$. Clearly, $\text{res}_{J_p}^J \tilde{\varphi} \sim \varphi$ so that $\text{res}_{J_p}^J$ is surjective. $\square$

For each prime $p$, we may apply proposition 3.1 to the component for $p$ in (3.1) and find that $H^1(J, N_p) \cong \text{inv}_J H^1(J_p, N_p) \cong \text{inv}_J H^1(J_p, N)$ for some $J_p \in \text{Syl}_p(J)$. In particular, it follows that:

**PROPOSITION 3.2.** Given a group $J$ acting on a nilpotent group $N$ via automorphisms so that $N \times J$ is supersoluble, the restriction maps $\text{res}_{J_p}^J$ induce an isomorphism of pointed sets $H^1(J, N) \cong \bigoplus_{p \in D} \text{inv}_J H^1(J_p, N)$ where $D$ denotes the set of prime divisors of $|J|$ and $J_p \in \text{Syl}_p(J)$ for each $p \in D$.

We are now prepared to provide a proof of lemma 1.4.

**Proof of lemma 1.4.** In a supersoluble group $G$, suppose $J$ and $J'$ are locally conjugate complements of a normal nilpotent subgroup $N$. As in lemma 1.1, we have for each prime $p$ that some $J_p \in \text{Syl}_p(J)$ and $J'_p \in \text{Syl}_p(J')$ are conjugate by an element of $N$. Let $\varphi' \in Z^1(J, N)$ denote the map corresponding to $J'$. As the isomorphism in proposition 3.2 is induced by restriction maps, it takes the identity $1 \in H^1(J, N)$ to $\bigoplus_{p \in D} 1|_{J_p}$. Thus, as $\varphi'|_{J_p} \sim 1|_{J_p}$ for each $p \in D$, we may apply proposition 3.2 to conclude $\varphi' \sim 1$ so that $J$ and $J'$ are conjugate. $\square$

We now use lemma 1.4 to show:
**PROPOSITION 3.3.** Let $H$ be a subgroup of some supersoluble $G \cong N \times J$ where $N$ is nilpotent. If for each prime $p$, $H$ contains a conjugate of some $S \in \mathrm{Syl}_p(J)$, then $H$ contains a conjugate of $J$ and so splits over $N \cap H$.

*Proof.* The hypotheses imply that $H$ supplements $N$ in $G$. We induct on the order of $G$. If $N$ is trivial or if $H$ is a $p$-group, the conclusion follows immediately. If multiple primes divide $|N|$, then for some prime $p$, $HN_p$ must be a strict subgroup of $G$ for $N_p \in \mathrm{Syl}_p(N)$; otherwise $H$ would contain a Sylow subgroup of $G$ for each prime and we would have $H = G$. Let $p$ be such a prime. Induction in $G/N_p$ implies $J^g \le HN_p$ for some $g \in G$. Switching to a conjugate of $H$ if necessary, we may assume that $g$ is trivial and apply the inductive hypothesis in $HN_p$ to conclude $J^{g'} \le H$ for some $g' \in G$. We now proceed under the assumption that $N$ is a $q$-subgroup for some prime $q$.

Let $A \le N$ be a minimal normal subgroup of $G$; as $G$ is supersoluble, it will have prime order $q$. If $A \le H$, then in $G/A$, induction implies that $J^g A \le HA = H$ for some $g \in G$ so that $J^g \le H$.

Otherwise, $A \cap H$ is trivial. Without loss, $J_q \le H$ for some $J_q \in \mathrm{Syl}_q(J)$. In $G/A$, induction implies that a conjugate of $JA/A$ is contained in $HA/A$. Let $\bar{K}$ denote this conjugate. Switching to a different conjugate if necessary, we may assume that $J_q A/A \le \bar{K}$. Let $\varphi : h \mapsto hA/A$ denote the isomorphism from $H$ to $HA/A$ and consider $K = \varphi^{-1}(\bar{K})$. It follows that $J_q \le K$ and $|K| = |J|$ so that $K \le H$ complements $N$ in $G$. As $N$ is a $q$-group, a Sylow $p$-subgroup of $J$ will be conjugate to a Sylow $p$-subgroup of $K$ for primes $p \ne q$. Lemma 1.4 then implies that $J$ and $K \le H$ are conjugate in $G$. $\square$

We now prove theorem 1.5.

*Proof of theorem 1.5.* Given $J, N$, and $\Omega$ as described in the hypotheses of the theorem, let $G = N \times J$ denote the induced semidirect product and consider $G_\alpha$ for some $\alpha \in \Omega$. As $N$ acts transitively, $G = NG_\alpha$. For each prime $p$, the hypotheses of the theorem imply $(J_p)^{n_p} \le G_\alpha$ for some $J_p \in \mathrm{Syl}_p(J)$ and $n_p \in N$, so that proposition 3.3 implies $G_\alpha$ contains a conjugate of $J$, say $J^g$ for $g \in G$. It follows that $J$ fixes $\omega = g \cdot a$. $\square$

This in turn implies:

**COROLLARY 3.4.** Let $G$ be a supersoluble split extension over a nilpotent subgroup $N$. If for each prime $p$ there is a Sylow $p$-subgroup $S$ of $G$ such that any two complements of $S \cap N$ in $S$ are conjugate, then any two complements of $N$ in $G$ are conjugate.

*Proof.* Suppose arbitrary $J$ and $J'$ complement $N$ in $G$. Then $G$ acts on the cosets $\Omega = G/J'$ in such a way that we may apply theorem 1.5 to infer that $J$ fixes $gJ'$ for some $g \in G$. Consequently, $J$ and $J'$ are conjugate, and we may conclude. $\square$

**4. Concluding remarks**

In their paper, Losey and Stonehewer exhibited a soluble group $G \cong N \times J$ with $N$ nilpotent and $J$ supersoluble and a second complement $J'$ to $N$ in $G$ such that
$J$ and $J'$ are locally conjugate but not conjugate [7]. Thus, lemma 1.4 cannot be extended to supersoluble complements of a normal nilpotent subgroup in a soluble group.

## Acknowledgements

The author thanks Elizabeth Crites, the editor Alex Bartel, and an anonymous reviewer for thoughtful and detailed feedback on the manuscript.

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