In this entry we prove that A_n is simple when n >= 5. We then use this result to prove that there is no generalization of “even” and “odd” permutation to three or more classes – i.e. there is no class of homomorphisms into Z_n, where n > 2.
In this entry, we bring together ideas we’ve been developing about permutations, their sign, cycles, and sorting as well as the run time complexity of the algorithms which compute them.
In the previous few entries we’ve been discussing quick_sort and analyzing the run-time complexity of recursive algorithms. We’re going to apply what we’ve learned so far to finding the median of an array in O(n) time. Then we’re going to see how that can be added to quick_sort to guarantee that it finishes in O(nlog_2(n)) time.
In the previous entry (Sorting, Random Permutations, and little bit of Probability), we introduced quick_sort, gave a version of it in C++ and started to analyze how many steps it takes to sort a vector of floating point numbers. In this entry, we continue that analysis and prove some results that will help us get a feel for other recursive algorithms.
We’re continuing our proofs of Sylow’s Theorems, and also continuing our efforts to count orbits, and determine the size of orbits created by the permutation groups…
In the previous entry, we gave a laborious proof that a group with an even number of elements has to have an element of order 2. We can do much better than that. We’ll show how by proving some famous theorems that were originally due to Sylow.
We continue our discussion of groups and permutations, breaking down a permutation into disjoint cycles, finding a proper subgroup of even permutations in a group of order 2(2k + 1), and discussing properties of the orbits, stabilizers, fixes, etc…
We’ve already introduced some of the topics of Group Theory. In this entry we’ll talk about Lagrange’s Theorem, various morphisms, and try to tie it back to our ongoing discussion of permutations. We’ll also give some examples of groups…
