Growth sequence of free product of alternating groups
Ahmad Erfanian · International Journal of Contemporary Mathematical Sciences · 2007
For a finitely generated group G, we denote G n as the direct product of n copies of G. The growth sequence of G is the sequence {d(G n )}n≥1, where d(G n ) is the minimum number of generators of G n . In this paper, we investigate the growth sequence of G, when G is the free product of alternating groups. In fact, we prove that d � (An ∗ Am) h(2,An)h(k,Am) � ≤ k +2 , for all n, m ≥ 5 and k ≥ 2, where h(2,An) is the maximum number t such that d(A t ) = 2 and similarly, h(k,Am) is the maximum number s such that d(A s )= k. Moreover, we will consider the case k = 2 and prove that d((An ∗ Am) t ) = 4, for all 1 ≤ t ≤ h(2,An)h(2,Am) and n,m ≥ 5. We have also confirmed the above results by several examples in find section.