Abstract. Let G be a finite group, let π(G) be the set of prime divisors of |G| and let Γ(G) be the prime graph of G. This graph has vertex set π(G), and two vertices r and s are adjacent if and only if G contains an element of order rs.

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Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$ , which is the maximum number of fixed points of an involution. In this paper, we study the involution fixity of almost simple primitive groups whose socle $T$ is an alternating or sporadic group; our main result classifies the groups of this form with ${\rm ifix}(T) \leqslant n^{4/9}$ . This builds on earlier work of Burness and Thomas, who studied the case where $T$ is an exceptional group of Lie type, and it strengthens the bound ${\rm ifix}(T) > n^{1/6}$ (with prescribed exceptions), which was proved by Liebeck and Shalev in 2015. A similar result for classical groups will be established in a sequel.

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In this paper we investigate the following general problem. Let G be a group and let i(G) be a property of G. Is there an integer d such that G contains a d-generated subgroup H with i(H) = i(G)? Here we consider the case where G is a profinite group and H is a closed subgroup, extending earlier work of Lucchini and others on finite groups. For example, we prove that d = 3 if i(G) is the prime graph of G, which is best possible, and we show that d = 2 if i(G) is the exponent of a finitely generated prosupersolvable group G.