# Comptes Rendus
Mathématique

Hamid Mousavi

Finite groups with Quaternion Sylow subgroup

Volume 358, issue 9-10 (2020), p. 1097-1099

Published online: 5 January 2021

https://doi.org/10.5802/crmath.131

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Group Theory / Théorie des groupes

# Finite groups with Quaternion Sylow
subgroup

Hamid Mousavi<sup>a</sup>

<sup>a</sup> Department of Mathematical Sciences, University of Tabriz, P.O.Box 51666-16471, Tabriz, Iran

E-mail: hmousavi@tabrizu.ac.ir

**Abstract.** In this paper we show that a finite group $G$ with Quaternion Sylow 2-subgroup is 2-nilpotent if, either $3 \nmid |G|$ or $G$ is solvable and the order of its Sylow 2-subgroup is strictly greater than 16.

**Mathematical subject classification (2010).** 20D99, 20E45.

Manuscript received 7th October 2020, revised 11th October 2020 and 13th October 2020, accepted 13th October 2020.

## 1. Introduction

A 2-group of order $2^n$, with a cyclic maximal subgroup is isomorphic to one of the following group:

(i) Cyclic group $\mathbb{Z}_{2^n}$ and abelian group $\mathbb{Z}_{2^{n-1}} \times \mathbb{Z}_2$.

(ii) Dihedral group $D_{2^n}$.

(iii) Semi-dihedral group $SD_{2^n}$.

(iv) Modular group $M(2^n)$.

(v) Quaternion group $Q_{2^n}$.

Let $G$ be a group with a Sylow 2-subgroup $S$ of type (i). Then $G$ is 2-nilpotent, since Aut($S$) is a 2-group and so $N_G(S) = \mathscr{C}_G(S)$. In [5], D. Gorenstein and J. Walter gave the characterization of finite groups with dihedral Sylow 2-subgroups. They proved that, a finite group $G$ with a dihedral Sylow 2-subgroup is 2-nilpotent if $G$ contains a normal subgroup of index 4 (see [5, Lemma 2.1(iii)]). In [8, 9], W. Wong obtained the structure of finite groups whose Sylow 2-subgroups are semi-dihedral or modular 2-group. In the latter case $G$ has a normal 2-complement (see [8, Theorem 1]), so a modular 2-group can not be a Sylow 2-subgroup of a non-solvable group. In the semi-dihedral case, $G$ is 2-nilpotent if $\mathrm{foc}(S) = S'$, where $S$ denotes a Sylow 2-subgroup of $G$ and $\mathrm{foc}(S)$, the focal subgroup of $S$ [8, Theorem 2(i)].

In [1], R. Brauer and M. Suzuki proved that any group with a Quaternion Sylow 2-subgroup is not simple.
Now this question seems to be natural: when a group $G$ with a Quaternion Sylow 2-subgroup is 2-nilpotent?

In this paper we obtain sufficient conditions for a group $G$ with a Quaternion (ordinary or generalized) Sylow 2-subgroup to be 2-nilpotent.

Our notations are standard and can be found in [6].

**Main Theorem.** Let $G$ be a finite group with a Quaternion (ordinary or generalized) Sylow 2-subgroup. Then $G$ is 2-nilpotent if:

(i) either $3 \nmid |G|$;

(ii) or $G$ is solvable with a Sylow 2-subgroup of order strictly greater than 16.

**Proof.** Assume that $Q \cong Q_{2^n}$ is a Sylow 2-subgroup of $G$. If $Q \trianglelefteq G$ then $G = QN$, where $N$ is a complement of $Q$. Now $N$ acts trivially on $Q$, since Aut$(Q_{2^n})$ is a 2-group unless for $n=3$, in the latter case Aut$(Q_8) \cong S_4$ and $3 \nmid |G|$, again $N$ acts trivially on $Q$. Hence $N \trianglelefteq G$ and $G \cong Q \times N$. Therefore we can assume that $Q \not\trianglelefteq G$.

(i). By [1], $G$ is not simple. Assume that $G$ is a minimal counterexample and $M$ is a maximal normal subgroup of $G$. If $|M|$ is odd, then $G/M$ is a simple group with a Quaternion Sylow subgroup which is a contradiction, so $2 \mid |M|$. By assumption $M \neq Q$. Suppose that $M \leq Q$, as $G/M$ is a non-abelian simple group and $3 \nmid |G/M|$, as the Suzuki groups are the only non-abelian simple groups which 3 does not divides its order [4], then $G/M$ is a Suzuki group with $Q/M$ as its Sylow 2-subgroup, which is a contradiction for $Q/M$ is either cyclic or dihedral but a Sylow 2-subgroup of a Suzuki group is of exponent 4 and order greater than or equal 64, (see [7, Lemma 1.6 (4)] and [2, Lemma 1 & Proposition 3]).

Therefore either $Q \leq M$, then $M$ has a normal 2-complement, for $|M| < |G|$, or $M \cap Q$ is a proper subgroup of $Q$ which is cyclic or $Q_{2^{n-1}}$, again $M$ has a normal 2-complement. Thus in either case $M$ has a normal 2-complement $M_1$ which is normal in $G$. Now $G/M_1$ has a Quaternion Sylow 2-subgroup. By the choice of $G$, $G \neq QM_1$ so $G/M_1$ has a normal 2-complement $N/M_1$. Hence $G = NQ$ which is a contradiction. Therefore $G$ is 2-nilpotent.

(ii). Proof by induction on $|G|$. Assume that $G$ is solvable and $n \geq 5$. Obviously $Q \neq G'$. If $Q \leq G'$, as $Q \not\trianglelefteq G'$, then by induction $G'$ is 2-nilpotent with a normal 2-complement $M$ which is normal in $G$. Now $G/M$ has a normal Sylow 2-subgroup so is 2-nilpotent with a normal 2-complement $N/M$. Obviously $N$ is normal a 2-complement of $G$. So assume that $G' \cap Q$ is a proper subgroup of $Q$, then either $G' \cap Q$ is cyclic which again implies that $G'$ is 2-nilpotent and we are done, or $G' \cap Q \cong Q_{2^{n-1}}$ and in this case $G'$ is 2-nilpotent unless $n=5$, $3 \mid |G'|$ and $G' \cap Q$ is non-cyclic and a non-normal subgroup of $G'$. In the latter case, $Q_1 = G' \cap Q \cong Q_{16}$ and $G^{(\ell)} \neq Q_1$, for all $\ell \geq 2$. By solvability for some $\ell$, $G^{(\ell)} \cap Q_1$ is a proper subgroup of $Q_1$. We can assume that $\ell$ is the smallest number such that $G^{(\ell)} \cap Q_1 \neq Q_1$, hence $G^{(\ell)} \cap Q_1$ is cyclic, since $G^{(\ell)} \cap Q_1 \trianglelefteq Q$. If $G^{(\ell)} \leq Q_1$, then $G^{(\ell-1)}/G^{(\ell)}$ is abelian with a normal 2-complement $M/G^{(\ell)}$, thus $MG^{(\ell)} \trianglelefteq G^{(\ell-1)}$ and so $M \trianglelefteq G^{(\ell-1)}$, for $G^{(\ell-1)} = MQ_1$. This implies that $M \trianglelefteq G$ and so $G/M$ is 2-nilpotent by induction. Therefore $G$ is 2-nilpotent. Otherwise $G^{(\ell)}$ is 2-nilpotent with a normal 2-complement $M$ which is normal in $G$, again $G/M$ is 2-nilpotent by induction and we are done. $\square$

**Corollary 1.** Let $G$ be a finite group with Quaternion (ordinary or generalized) Sylow 2-subgroup. If a Sylow 3-subgroup of $G$ is normal, then $G$ is 2-nilpotent.

**Proof.** Assume that $P$ is the Sylow 3-subgroup of $G$. As $3 \nmid |G/P|$, $G/P$ is 2-nilpotent by the main Theorem. So $G$ is 2-nilpotent. $\square$

**Corollary 2.** Let $G$ be a finite group with Quaternion (ordinary or generalized) Sylow 2-subgroup such that $3 \nmid |G|$, then $G$ is solvable.
**Remark 3.** In the above lemma for $n = 3$ and $4$ where $3 \mid |G|$, there exist many examples for which $G$ is not 2-nilpotent. The smallest order of such groups is a group of order 48. Assume that $G := \text{SmallGroup}(48,28)$ the small group of GAP library [3], in this group $G' \cong \text{SL}(2,3)$ and a Sylow 2-subgroup of $G$ is non-normal, obviously $G/G' \cong \mathbb{Z}_2$. For another example we consider $G \cong \mathbb{Z}_5 \times \text{SL}(2,3)$. In this case $G' \cong \text{Q}_8$ and $G \cong (\mathbb{Z}_7 \times \mathbb{Z}_7) \times \text{SL}(2,3)$, where $\text{Q}_8$ acts irreducibly on $\mathbb{Z}_7 \times \mathbb{Z}_7$, in this case $G' \cong (\mathbb{Z}_7 \times \mathbb{Z}_7) \times \text{Q}_8$. For non-solvable case let $G := \text{SmallGroup}(672,1045)$, in this group $G' \cong \text{SL}(2,7)$ and a Sylow 2-subgroup of $G$ is a Quaternion group of order 32.

Acknowledgment

The author would like to thank professor Stefan Kohl for providing useful information about simple groups, especially the Suzuki groups.

References

[1] R. Brauer, M. Suzuki, "On finite groups of even order whose 2-Sylow group is a Quaternion group", *Proc. Natl. Acad. Sci. USA* **45** (1959), p. 1757-1759.

[2] M. J. Collins, "The characterisation of the Suzuki groups by their Sylow 2-subgroups", *Math. Z.* **123** (1971), p. 32-48.

[3] The GAP Group, "GAP - Groups, Algorithms, and Programming, Version 4.11.0", 2020, http://www.gap-system.org.

[4] G. Glauberman, *Factorizations in local subgroups of finite groups*, Regional Conference Series in Mathematics, vol. 33, American Mathematical Society, 1977.

[5] D. Gorenstein, J. H. Walter, "The characterization of finite groups with dihedral Sylow 2-subgroups I., II., III", *J. Algebra* **2** (1965), p. 85-151; 218-270; 334-393.

[6] I. M. Isaacs, *Finite group theory*, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, 2008.

[7] S. Kohl, "Counting the orbits on finite simple groups under the action of the automorphism group - Suzuki groups vs. linear groups", *Commun. Algebra* **30** (2002), no. 7, p. 3515-3532.

[8] W. J. Wong, "On finite groups whose 2-Sylow subgroups have cyclic subgroups of index 2", *J. Aust. Math. Soc.* **4** (1964), p. 90-112.

[9] ——, "On Finite Groups with Semi-Dihedral Sylow 2-Subgroups", *J. Algebra* **4** (1966), p. 52-63.