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Let G be a finite group. An element x in G is a real element if x isconjugate to its inverse in G. For x in G, the conjugacy class x^G is said tobe a real conjugacy class if every element of x^G is real. We show that if 4divides no real conjugacy class size of a finite group G, then G is solvable.We also study the structure of such groups in detail. This generalizes severalresults in the literature.

Groups with some arithmetic conditions on real class sizes