Squaring a conjugacy class and cosets of normal subgroups. (Q2790912)

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.NEWLINENEWLINE 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.NEWLINENEWLINE 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.NEWLINENEWLINE 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.