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چکیده
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In this paper we estimate the diameter of a direct power of alternating groups Ak for k ≥ 4. We show that there exists a generating set of minimum size for An 4 , for which the diameter of An 4 is O(n). For k ≥ 5, we show that there exists a generating set of minimum size for A2 k , for which the diameter of A2 k is at most O(ke(c+1)(log k) 4 log log k) , for an absolute constant c > 0. Finally for 1 ≤ n ≤ 8, we provide generating sets of size two for An 5 and we show that the diameter of An 5 with respect to those generating sets is O(n). These results leads us to the sense that the best upper bound known for the diameter of the direct power of non-abelian simple groups (specially alternating groups), i.e. O(n 3 ) [5], may be improved to O(n). Furthermore, these results are more pieces of evidence for a conjecture which has been presented in [9] in 2015.
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