The p-width of the alternating groups

Abstract

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.

The involution width of finite simple groups

“…We therefore restrict our attention to odd primes. But we note that this has also appeared in the literature before: the p-width of the alternating groups is studied by this author in the unpublished note [21], and Theorem 2.1 is proved there by direct combinatorial methods. Here we present a greatly simplified proof of Theorem 2.1 using existing work of Dvir [6].…”