On Strong Reality of Finite Simple Groups
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On Strong Reality of Finite Simple Groups S. G. KOLESNIKOV and JA. N. NUZHIN Krasnoyarsk State Technical University, ul. Kirenskogo 26, 660074 Krasnoyarsk, Russia. e-mail: [email protected] Abstract. In this paper we shall consider the following problem. Which finite simple groups have the property: any element is inverted by an involution? Mathematics Subject Classification (2000): 20D06. Key words: finite simple group, strongly real element, complex character.
We shall call a group G strongly real if any its element g = 1 is strongly real, i.e., if g and g −1 are conjugated by an involution of G. Here the following problem is considered. Which finite simple groups are strongly real? Our main result is the following theorem. THEOREM 1. (a) Among 26 sporadic groups only two Janko’s groups J1 and J2 are strongly real. (b) The following simple groups of Lie type, defined over the field GF(q), are not strongly real: (b1) (b2) (b3) (b4)
For any q the groups of type Al (l 2), 2 Al (l 2), 2 B2 , 2 G2 . For q ≡ 3 (mod 4) and l 1 the groups of type Cl , B4l+1 , B4l+2 , D4l+2 , E7 . For odd q > 3 and l 3 the groups of type D2l−1 . For q > 3 the groups of type E6 .
Earlier it was established that the symplectic group PSp2n (q) is strongly real for q = 2m [4, 5] and q ≡ 1 (mod 4) [6]. It is also proved that the alternating group An (n 5) is strongly real if and only if n = 5, 6, 10, 14 [1]. Thus, the problem is solved for sporadic, alternating, special linear, symplectic and unitary groups, and for Suzuki groups and Ree groups also. This research was supported by the Russian Foundation for Basic Research.
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S. G. KOLESNIKOV AND JA. N. NUZHIN
1. Some Properties of Strongly Real Groups If G is a strongly real group then by definition for any its element g = 1 there exists an involution i ∈ G such that igi = g −1 . Hence, if g = i, then g is the product of two involutions i and ig. A group G is called bireflectional (2-reflectional) if every element of G is a product of two involutions. Such groups play an important role in geometry. At present it is proved, that some classical groups are bireflectional (see [1, 4–6]). In a finite simple group any involution belongs to some Klein group. Therefore, a finite simple group is strongly real if and only if it is bireflectional. Note also, that in 1999 A. I. Sozutov took down in ‘The Kourovka Notebook’ (as known) the following problem: To describe finite simple groups, in which every element is a product of two involutions [7, Problem 14.82]. It is obvious that elementary Abelian 2-groups and dihedral groups are strongly real, but a dihedral group is bireflectional if and only if its kernel has an even order. The following properties of strongly real groups arise easily from the definition. PROPOSITION 2. The class of strongly real groups is closed under taking of direct products and quotients. It is well known that an element of a finite group is conjugate to its inverse if and only if values of all complex characters on this element a
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