Corrigendum to “Intransitive Permutation Groups

Zeinab Foruzanfar

Corrigendum to “Intransitive Permutation Groups

Zeinab Foruzanfar

2022, Mathematical Reports

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This corrigendum addresses an omission in the previous work concerning intransitive permutation groups with bounded movement of maximum degree. It corrects part (II) of Theorem 1.1 by introducing two cases that were previously overlooked, providing an example to illustrate the case that satisfies the conditions of the theorem. The results show that the only group that fits the new criteria is a specific semidirect product, refining the theoretical understanding of the structure of such groups and correcting the classification of permutation groups.

CORRIGENDUM TO “INTRANSITIVE PERMUTATION GROUPS WITH BOUNDED MOVEMENT HAVING MAXIMUM DEGREE” MEHDI REZAEI, MEHDI ALAEIYAN, and ZEINAB FORUZANFAR Communicated by Sorin Dăscălescu AMS 2020 Subject Classification: 20B05. Key words: intransitive permutation groups, orbits, movement. In this note, we point out a mistake in part (II) of Theorem 1.1 of [1] and correct it here. In [1], the authors investigated the structure of intransitive permutation groups with bounded movement having maximum degree. In part (II) of Theorem 1.1, they classified all permutation groups with bounded movement m having maximum degree n = |Ω| = 3m + 1. Unfortunately, they missed a group satisfying in this condition. In fact, at the bottom of page 6, they omitted the cases (d) and (g) of the cases (a) − (o), i.e., the cases r3 = s = 1, r2 = r3′ = r4 = 0 and r3′ = s = 1, r2 = r3 = r4 = 0. In this note, we consider these two cases and provide an example satisfying in the first case which was not mentioned in part (II) of Theorem 1.1. Also, we note that the cases (h), (i), (j), (m), (n) (o), were not necessary to check. Because 3 is the least odd prime dividing the order of G and it is not possible to have r3 = r3′ = 0. As it was checked in [1], the cases (a), (b), (e), (k), (l) do not arise by the equality n = |Ω| = 3r3 + 3r3′ + 2r2 + 4r4 + |Σ5 | = 3m + 1. So, only the following four cases remain: (I) r3 = r4 = 1, r3′ = r2 = s = 0, (II) r3 = s = 1, r2 = r3′ = r4 = 0, (III) r3′ = r4 = 1, r3 = r2 = s = 0, (IV) r3′ = s = 1, r2 = r3 = r4 = 0, The cases (I) and (III) were verified in [1]. Below, we give an example which satisfies the case (II). MATH. REPORTS 25(75) (2023), 3, 451–453 doi: 10.59277/mrar.2023.25.75.3.451 452 M. Rezaei, M. Alaeiyan, and Z. Foruzanfar 2 Example 1. Let G1 := Z7 and G2 := Z3 be permutation groups on the sets Ω1 = {1, 2, 3, 4, 5, 6, 7} and Ω2 = {1′ , 2′ , 3′ }, respectively, where Z7 ∼ = ⟨(1 · · · 7)⟩ ′ ′ ′ ∼ and Z3 = ⟨(1 2 3 )(2 3 5)(4 7 6)⟩. Set Ω := Ω1 ∪ Ω2 . Then G := G1 ⋊ G2 is a permutation group on Ω which has t = 2 orbits, and since each non-identity element of G has three cycles of length 3 or one cycle of length 7, so m = move(G) = 3. It follows that n = 3m + 1 = 10. PNow we consider the cases (II) and (IV). For the sake of simplicity, we set 5 := ∆. So, we have n = 3m + 1 = 3 + |∆|. Suppose that the transitive constituent G∆ of G has movement m′ . Let p ≥ 5 be the least odd prime dividing |G∆ |. Since we are trying to classify permutation groups with bounded movement having maximum degree, by ′p ∆ Lemma 1.1 of [2], we can consider the maximum degree ⌊ 2m p−1 ⌋ for G . Then by checking Theorem 1.2 of [2], one can see that the cases (1) and (2) do not happen. So, G∆ is a p-group of order pa on the set ∆ of size pa with movement a−1 m′ = p 2(p−1) , for some a ≥ 1. Therefore, case (IV) does not arise. Now inequality m ≥ m′ implies that p 3+2 ≥ p 2(p−1) . But this only happens when p = 7, a = 1, m = m′ = 3 and n = 10, which implies that the only group satisfying in this case is the group mentioned in Example 1. a a−1 Now Theorem 1.1 of [1] can be corrected as follows: Theorem 2. Let G be a permutation group on a set Ω with t, t ≥ 2, orbits which have no fixed points in Ω. Suppose further that m is a positive integer such that move(G) = m and n = 3m + t − 1. Then I) n ≤ ⌊ 9m−3 2 ⌋, and the equality holds if and only if G is one of the following: (a) G is an elementary abelian 3-group and all its orbits have length 3. (b) G is the semidirect product of Z22 and Z3 with normal subgroup Z22 . II) Let n = 3m + 1, which is the maximum bound for t = 2. Then either G is the group mentioned in the case (b) or G is the semidirect product of Z7 and Z3 with normal subgroup Z7 . Acknowledgments. The authors would like to thank the referees for carefully reading the manuscript and for giving constructive comments. 3 Corrigendum to “Intransitive permutation groups with bounded movement” 453 REFERENCES [1] M. Alaeiyan and M. Rezaei, Intransitive permutation groups with bounded movement having maximum degree. Math. Reports (Bucur.) 13(63) (2011), 2, 109-115. [2] A. Hassani, M. Khayaty (Alaeiyan), E.I. Khukhro, and C. E. Praeger, Transitive permutation groups with bounded movement having maximal degree. J. Algebra 214 (1999), 317-337. Received April 10, 2020 Buein Zahra Technical University Department of Mathematics Buein Zahra, Qazvin, Iran [email protected], m [email protected] Iran University of Science and Technology Department of Mathematics Narmak, Tehran, 16844 Iran [email protected] Buein Zahra Technical University Department of Mathematics Buein Zahra, Qazvin, Iran [email protected], z [email protected]

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