A New Characterization of Sporadic Simple Group M₂₂ via Its Vanishing Elements

For a finite group G, a vanishing element of G is an element g ∈ G such that χ(g) = 0 for some irreducible complex character χ of G. Denote by πe(G) the set of orders of elements in G and by Vo(G) the set of the orders of vanishing elements of G. A group M is said to be recognizable if every finite group N with πe(M) = πe(N) is isomorphic to M, and V-recognizable if every finite group N with Vo(M) = Vo(N) is isomorphic to M. This paper investigates the relationship between recognizable and V-recognizable groups. As an application, we show that G ≅ M₂₂ if and only if Vo(G) = Vo(M₂₂). Hence we give a partial positive answer to [arXiv:1401.0300v12, Problem 19.30].