## Correction to: Monatshefte für Mathematik https://doi.org/10.1007/s00605-018-1256-x

**Abstract**

In this Corrigendum we correct a missed case in the statement of Theorem 2.4 and a subsequent mistake in the proof of the main result in “A coprime action version of a solubility criterion of Deskins”, Monatsh. Math. **188**, 461–466 (2019).

**Keywords** Soluble groups \(\cdot \) Maximal subgroups \(\cdot \) Coprime action \(\cdot \) Group action on groups

**Mathematics Subject Classification** 20D20 \(\cdot \) 20D15

The main result of [1] is a coprime action version of a theorem of B. Huppert: If a finite group *G* has a maximal subgroup that is nilpotent with Sylow 2-subgroup of nilpotency class at most 2, then *G* is soluble (Satz IV.7.4 of [4]). This theorem is the completion of previous results by Huppert [5], J.G. Thompson [8], W.E. Deskins [2] and Z. Janko [6]. Professor M.D. Pérez Ramos noticed and informed us that there are some mistakes and inaccuracies in the last part of the proof of the main theorem of [1]. Thus the goal of this note is to correct them.

First, the proof of the main theorem uses two classification theorems due to Kondrat’ev [7] and to Gilman and Gorenstein [3], respectively. However, there is one simple group missed in the statement of Theorem 2.4 in [1], which joins both classifications. The correct statement is the following.

### Theorem 2.4

Let *G* be a finite non-abelian simple group and *P* a Sylow 2-subgroup of *G*. If \(\mathbf{N}_G(P)=P\) and *P* has class at most 2, then \(G\cong \mathrm{PSL}(2,q)\), where \(q\equiv 7,9\) (mod 16) or \(G\cong A_7\).

### Proof

This is a consequence of combining the main result of [7] and Theorems 7.1 and 7.4 of [3]. \(\square \)

As a consequence of this correction, several modifications in the proof of Step 4 of the main theorem of [1] are necessary. Furthermore, in lines 28–29, page 465 it is claimed that the subgroup *K* is normalised by an element of order 3 lying in *S*. This is not true. For the reader’s convenience we rewrite the whole proof of Step 4.

*Proof of Step 4* Let *N* be a minimal *A*-invariant normal subgroup of *G*. We can assume that *N* is not soluble; otherwise by Step 1, *N* is not contained in *M*, and by maximality we obtain \(NM=G\). As a consequence, *G* would be soluble and the proof is finished. Therefore, we can write \(N=S_1\times \ldots \times S_n\) where \(S_i\) are isomorphic non-abelian simple groups (possibly \(n=1\)). Put \(S=S_1\). Notice that *A* permutes the \(S_i's\), but not necessarily this action is transitive. Let \(B=\mathbf{N}_A(S)\) and let *T* be a transversal of *B* in *A*. On the other hand, since *M* is maximal in *G*, we have \(\mathbf{N}_G(M\cap N)=M\), so in particular \(\mathbf{N}_N(M\cap N)=M\cap N\). Further, as \(M\cap N\) is a Sylow 2-subgroup of *N*, we have \(M\cap N=M\cap S \times \ldots \times M\cap S_n\), so we conclude that \(M\cap S\) is self-normalising in *S*. Also, it has nilpotency class exactly 2 by Lemma 2.1 and Step 3. Then by applying Theorem 2.4, we obtain \(S\cong \) PSL(2, *q*) with \(q\equiv 7,9\) (mod 16) or \(S\cong A_7\). We distinguish separately these cases.

Assume first that \(q\equiv 9 \) (mod 16), with \(q>9\). Then we can certainly choose an odd prime \(r\mid (q-1)/2\) and *R* to be a *B*-invariant Sylow *r*-subgroup of *S*. By Lemma 2.5(3), we know that \(|\mathbf{N}_S(R)|= q-1\), so \(\mathbf{N}_S(R)\) has odd index in *S* and contains properly a Sylow 2-subgroup of *S*. Analogously, if \(q\equiv 7 \) (mod 16), with \(q>7\), there exists an odd prime \(r\mid (q+1)/2\) and we take *R* to be a *B*-invariant Sylow *r*-subgroup of *S*. Again by Lemma 2.5(2), we know that \(|\mathbf{N}_S(R)|= (q+1)\), so \(\mathbf{N}_S(R)\) has odd index in *S* and hence, it contains properly a Sylow 2-subgroup of *S*. In both cases, we put \(R_1=\prod _{t\in T}R^t\), which is an *A*-invariant Sylow *r*-subgroup of \(\prod _{t\in T} S_1^t\). We can argue similarly to construct an *A*-invariant Sylow *r*-subgroup for each orbit of the action of *A* on the \(S_i's\). Hence, we can construct \(R_0=R_1 \times \ldots \times R_t\), where *t* denotes the number of orbits of *A* on the \(S_i's\), and this is certainly an *A*-invariant Sylow 2-subgroup of *N*. We conclude that \(|N:\mathbf{N}_N(R_0)|=|S:\mathbf{N}_S(R)|^n\) is odd too. Now, by the Frattini argument, \(G=N\mathbf{N}_G(R_0)\) and thus, \(|G:\mathbf{N}_G(R_0)|=|N: \mathbf{N}_N(R_0)|\). We conclude that \(\mathbf{N}_G(R_0)\) properly contains an *A*-invariant Sylow 2-subgroup of *G*, contradicting the maximality of *M*.

Finally, suppose that \(S\cong \mathrm{PSL}(2,9), \mathrm{PSL}(2,7)\) or \(A_7\). In all cases, the Sylow 2-subgroups of *S* are dihedral groups of order 8. Now, \(M\cap N\) is an *A*-invariant Sylow 2-subgroup of *N*, which is the direct product of *n* copies of a dihedral group, say *D*, of *S*. As *M* has nilpotence class two, then \([M, M\cap N]\le M'\le \mathbf{Z}(M)\), and since \(M\cap N\unlhd M\) it follows that \([M, M\cap N]\le \mathbf{Z}(M) \cap (M\cap N) \le \mathbf{Z}(M\cap N)\). This implies that every subgroup of \(M\cap N\) containing \(\mathbf{Z}(M\cap N)\) must be normal in *M*. We will use this property later. Now let *K* be one of the two subgroups of *D* isomorphic to the 4-Klein group, which obviously satisfies \(\mathbf{Z}(D)\le K\) and set \(K^A=\langle K^a \mid a\in A\rangle \). By the coprime action hypothesis we have that |*A*| is odd, and then the fact that *D* has exactly two subgroups isomorphic to the 4-Klein group implies that for every \(a\in A\), either \(K^a=K\), or \(K^a\) lies in some other distinct copy of *S*. Furthermore, \(K^A\) is a direct product of certain copies of *K*, each of which lies in a different copy of *S*. Now, if the action of *A* on the \(S_i's\) is transitive, we will just consider the subgroup \(K^A\), but if the action is not transitive, then we proceed as follows. For each of the orbits of the action of *A* on the \(S_i\), we choose *j* with \(S_j\) in the orbit, and choose a 4-Klein subgroup \(K_j\le D_j\), where \(D_j\) is the corresponding isomorphic copy of *D* appearing in \(M\cap N\). Then we define the subgroup \(K_j^A\) similarly as \(K^A\). Set \(K_0\) to be the direct product of these subgroups, one for each orbit of the action of *A* on the \(S_i\). We can write \(K_0=K_1\times \ldots \times K_n\), where each \(K_i\) is a 4-Klein group lying in \(S_i\). By construction, \(K_0\) is trivially *A*-invariant and, moreover, \(K_0\unlhd M\), because \(\mathbf{Z}(M\cap N)\le K_0\le M\cap N\). By the above proved property, we get \(M\le \mathbf{N}_G(K_0)\), which is also *A*-invariant. Now \(\mathbf{N}_G(K_0)=M \mathbf{N}_N(K_0)\) and \(\mathbf{N}_N(K_0)=\prod _{i=1}^n\mathbf{N}_{S_i}(K_i)\). In fact, one can easily check that when \(S\cong \mathrm{PSL}(2,9)\) or \(\mathrm{PSL}(2,7)\) then \(\mathbf{N}_{S}(K)\cong S_4\), and when \(S\cong A_7\), then \(\mathbf{N}_{S}(K)\cong (A_4 \times C_3) < imes C_2\). In all cases we get a contradiction with the maximality of *M*.

### Remark

It is possible to give a simpler argument for the case \(S\cong A_7\) by using that \(A_7\) possesses a unique conjugacy class of \(\{2, 3\}\)-Hall subgroups. In this case, by Glauberman’s Lemma, there exists an *A*-invariant \(\{2,3\}\)-Hall subgroup of *N*, say *H*. Then the Frattini argument gives \(G=N\mathbf{N}_G(H)\), so \(|G:\mathbf{N}_G(H)|\) is a \(\{2, 3\}'\)-number. This implies that the *A*-invariant subgroup \(\mathbf{N}_G(H)\) properly contains an *A*-invariant Sylow 2-subgroup of *G*, contradicting the maximality of such Sylow 2-subgroup (Step 2).

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## Acknowledgements

We would like to thank M.D. Pérez-Ramos for many helpful conversations on the subject. The first author was partially supported by Ministerio de Ciencia, Innovación y Universidades, Proyecto PGC2018-096872-B-100 and also by Proyecto UJI-B2019-03. The second author was supported by the Nature Science Fund of Shandong Province (No. ZR2019MA044) and the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (No. 2018QZJ04).

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Beltrán, A., Shao, C. Correction to: A coprime action version of a solubility criterion of Deskins.
*Monatsh Math* (2020). https://doi.org/10.1007/s00605-020-01367-x

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