“Characterization of the groups PSL 5(2), PSL 6(2) and PSL 7(2)” by Mohammad Reza Darafsheh Footnote* and Ali Reza Moghaddamfar, Communications in Algebra ®, 29(1), 2001, pp. 465–475.
In Darafsheh and Moghaddamfar (Citation[2001]) we conjectured that the groups L n (2), n > 2, are characterizable by their order elements and this conjecture has been verfied for n = 5, 6 and 7. Although our proof for the case n = 5 is correct, but in cases n = 6 and 7 there is an error at the end of the proof of Theorems 3.2 and 3.3. The error occurs in both cases because the symmetric group S 8is not a 2-solvable group, and therefore Lemma 2.10 of Darafsheh and Moghaddamfar (Citation[2001]) is not applicable.
Our purpose of writing this short note is to establish the above results in order to provide a correction for Theorems 3.2 and 3.3 in Darafsheh and Moghaddamfar (Citation[2001]).
In fact the error occurred for the simple group L 6(2) was corrected in the following Lemma due to Shi and others which has been published in Shi et al. (Citation[2003]).
Lemma 1
Let S = L 6(2) and Nbe a non-trivial elementary Abelian 2-group. Then π e (N.S) ≠ π e (S), where N.Sdenotes an extension of the group Nby S.
Here, we will prove the following Lemma.
Lemma 2
Let Gbe a finite group with an elementary Abelian normal 2-subgroup Nsuch that G/N ≅ L 7(2). If π e (G) = π e (L 7(2)) then N = 1.
Proof
Suppose false and N ≠ 1. We set L = L 7(2). Let H/N < Land H/N ≅ L 5(2). We use the 2-Brauer character Table of L 5(2) computed in Jansen et al. (Citation[1995]). We use notations used in page 172 of Jansen et al. (Citation[1995]) without any comments. By definition, b 7 + b 7** = −1, so, for every 2-Brauer character of L 5(2), we have
Let xbe an element of order 7 in Hand x¯ = xNbe the corresponding image of xin L 7(2). Then C = C N (x) ≠ 1, and C L (x¯) ≅ ⟨x¯⟩ × L 4(2) acts on C. If this action is trivial, then an element of order 5 from C L (x¯) centralize C, so Gcontains an element of order 2.5.7, which is a contradiction. If this action is non-trivial, then since L 4(2) contains a Frobenius group of order 5.4, C.L 4(2) contains an element of order 8 which centralizes x, so Gcontains an element of order 7.8, which is a contradiction. Therefore N = 1.
Finally we would like to mention a misprint concerning references in Darafsheh and Moghaddamfar (Citation[2001]) and the way they are quoted in the paper. In fact quotation for references 1–6 in Darafsheh and Moghaddamfar (Citation[2001]) must remain the same, but from 7 to 26 one must subtract 1 from each of the quoted numbers, because the last reference in Darafsheh and Moghaddamfar (Citation[2001]) is 25.
Acknowledgment
We are greatly indebted to Professor V.D. Mazurov for his help in the proof of Lemma 2.
Notes
*Correspondence: Mohammad Reza Darafsheh, Department of Mathematics and Computer Science, Faculty of Science, University of Tehran, Tehran, Iran; E-mail: [email protected].
References
- Darafsheh , M. R. and Moghaddamfar , A. R. 2001 . A characterization of the groups PSL 5(2), PSL 6(2) and PSL 7(2) . Comm. Algebra , 29 ( 1 ) : 465 – 475 .
- Shi , W. J. , Wang , L. H. and Wang , S. H. 2003 . The pure quantitative characterization of linear groups over the binary field (Chinese) . Chinese Annals of Mathematics (Ser. A) , 24 to appear
- Jansen , C. , Lux , K. , Parker , R. and Wilson , R. 1995 . An Atlas of Brauer Characters Oxford University Press .
- *Correspondence: Mohammad Reza Darafsheh, Department of Mathematics and Computer Science, Faculty of Science, University of Tehran, Tehran, Iran; E-mail: [email protected].