{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def print_commutator_series(n, full=False):\n",
    "    G = SymmetricGroup(n)\n",
    "    G_series = [set(G)]\n",
    "    for i in range(20):\n",
    "        Gi = G_series[i]\n",
    "        if(full):\n",
    "            print(\"%s:\\n%s\\n\" % (i, Gi))\n",
    "        else:\n",
    "            print(\"%s :\\t %s\" % (i, len(Gi)))\n",
    "            \n",
    "        if len(Gi) == 1:\n",
    "            break\n",
    "\n",
    "        H = set()\n",
    "        for g in Gi:\n",
    "            for h in Gi:\n",
    "                H.add(g * h * g.inverse() * h.inverse())\n",
    "        G_series.append(H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print_commutator_series(4, full=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print_commutator_series(3, full=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print_commutator_series(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print_commutator_series(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print_commutator_series(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = Permutation([(1,2,3,4,5)])\n",
    "g.cycle_type() != [5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def find_Barrington_cycles(n):\n",
    "    G = SymmetricGroup(n)\n",
    "    for g in G:\n",
    "        if g.cycle_type() != [n]: continue\n",
    "        for h in G:        \n",
    "            if h.cycle_type() != [n]: continue\n",
    "\n",
    "            commutator = g * h * g.inverse() * h.inverse()\n",
    "            if commutator.cycle_type() == [n]:\n",
    "                return (g, h, commutator)\n",
    "\n",
    "print(find_Barrington_cycles(5))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = PermutationGroupElement([(1,2,3,4,5)])\n",
    "h = PermutationGroupElement([(1,3,5,4,2)])\n",
    "g * h * g.inverse() * h.inverse()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "SageMath 8.4",
   "language": "",
   "name": "sagemath"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.15"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}