Symmetric Groups
The symmetric group Sn can be generated by two elements, a 2-cycle and an n-cycle, so that it is a quotient group of F2. On the other hand, it is easy to show that the maximal order M(n) of an element in Sn satisfies
- log M(n) ≤ n/e
(Edmund Landau proved the more precise asymptotic estimate log M(n) ~ (n log n)1/2). In fact if the cycles in a cycle decomposition of a permutation have length N1, ..., Nk with N1 + ··· + Nk = n, then the order of the permutation divides the product N1 ···Nk, which in turn is bounded by (n/k)k, using the inequality of arithmetic and geometric means. On the other hand, (n/x)x is maximized when x=e. If F2 could be written as a product of m cyclic subgroups, then necessarily n! would have to be less than or equal to M(n)m for all n, contradicting Stirling's asymptotic formula.
Read more about this topic: Boundedly Generated Group, Free Groups Are Not Boundedly Generated
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