In this paper we measure how efficiently a finite simple group G is generated by its elements of order p, where p is a fixed prime. This measure, known as the p-width of G, is the minimal k ∈ N such that any g ∈ G can be written as a product of at most k elements of order p. Using primarily character theoretic methods, we sharply bound the p-width of some low rank families of Lie type groups, as well as the simple alternating and sporadic groups.
Let p be a fixed prime. For a finite group generated by elements of order p, the p-width is defined to be the minimal k ∈ N such that any group element can be written as a product of at most k elements of order p. Let An denote the alternating group of even permutations on n letters. We show that the p-width of An (n ≥ p) is at most 3. This result is sharp, as there are families of alternating groups with p-width precisely 3, for each prime p.
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