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
Famous quotes containing the word groups:
“Women over fifty already form one of the largest groups in the population structure of the western world. As long as they like themselves, they will not be an oppressed minority. In order to like themselves they must reject trivialization by others of who and what they are. A grown woman should not have to masquerade as a girl in order to remain in the land of the living.”
—Germaine Greer (b. 1939)