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Squaring a conjugacy class and cosets of normal subgroups. (English) Zbl 1346.20037

The product of two conjugacy classes of a finite group \(G\) is an invariant set under conjugacy and it may be again a conjugacy class. For instance, this happens for all classes in abelian groups, or when just one of both classes is a central element of \(G\), or even when both classes have coprime cardinality.
In this paper, the authors characterize the particular case in which the square of a single conjugacy class \(K=x^G\) of \(x\in G\) is a conjugacy class.
Theorem A claims that this occurs if and only if \(K=x[x,G]\) and \(C_G(x)=C_G(x^2)\), or equivalently, and by using the set of irreducible characters of \(G\), if and only if \(\chi(x)=0\) or \(|\chi(x)|=\chi(1)\) for every \(\chi\in\mathrm{Irr}(G)\), and \(C_G(x)=C_G(x^2)\). What is more relevant, under such hypothesis the normal subgroup \([x,G]\) is always solvable, and for proving this nice result the authors employ the Classification of Finite Simple Groups. Thus, the fact that the square of a conjugacy class is a conjugacy class provides a solvable normal subgroup in \(G\). This property agrees with the still open Arad and Herzog’s conjecture, which asserts that the product of two non-trivial conjugacy classes of a non-abelian finite simple group can never be a conjugacy class.
The other main result of the paper is Theorem B and concerns conjugacy and coclasses of a normal subgroup. It is inspired by the fact that the class \(K\) in Theorem A is really a coclass of a solvable normal subgroup. Theorem B establishes that if \(N\) is a normal subgroup of a finite group \(G\) and \(x\in G\) then: (a) if all elements of \(xN\) are \(G\)-conjugate, then \(N\) is solvable; (b) if all elements of \(xN\) are \(G\)-conjugate and \(x\) is a \(p\)-element for some prime \(p\), then \(N\) has a normal \(p\)-complement; (c) if all elements of \(xN\) have odd order, then \(N\) is solvable. – The proofs of (a) and (c) also require the Classification of Finite Simple Groups but (b) does not.

MSC:

20E45 Conjugacy classes for groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D06 Simple groups: alternating groups and groups of Lie type
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
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References:

[1] Arad, Z.; Herzog, M., Products of conjugacy classes in groups, Lecture Notes in Mathematics 1112, i+244 pp. (1985), Springer-Verlag, Berlin · Zbl 0561.20004 · doi:10.1007/BFb0072284
[2] Bianchi, Mariagrazia; Chillag, David; Lewis, Mark L.; Pacifici, Emanuele, Character degree graphs that are complete graphs, Proc. Amer. Math. Soc., 135, 3, 671-676 (electronic) (2007) · Zbl 1112.20006 · doi:10.1090/S0002-9939-06-08651-5
[3] Fried, Michael D.; Guralnick, Robert; Saxl, Jan, Schur covers and Carlitz’s conjecture, Israel J. Math., 82, 1-3, 157-225 (1993) · Zbl 0855.11063 · doi:10.1007/BF02808112
[4] Glauberman, George, Correspondences of characters for relatively prime operator groups., Canad. J. Math., 20, 1465-1488 (1968) · Zbl 0167.02602
[5] Guralnick, Robert M.; Malle, Gunter; Tiep, Pham Huu, Products of conjugacy classes in finite and algebraic simple groups, Adv. Math., 234, 618-652 (2013) · Zbl 1277.20029 · doi:10.1016/j.aim.2012.11.005
[6] Isaacs, I. M., Characters of solvable and symplectic groups, Amer. J. Math., 95, 594-635 (1973) · Zbl 0277.20008
[7] Isaacs, I. Martin, Character theory of finite groups, xii+310 pp. (2006), AMS Chelsea Publishing, Providence, RI · Zbl 1119.20005
[8] Ladisch, Frieder, Groups with anticentral elements, Comm. Algebra, 36, 8, 2883-2894 (2008) · Zbl 1182.20018 · doi:10.1080/00927870802108106
[9] Rowley, Peter, Finite groups admitting a fixed-point-free automorphism group, J. Algebra, 174, 2, 724-727 (1995) · Zbl 0835.20036 · doi:10.1006/jabr.1995.1148
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