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    "This is an ipython (jupyter) worksheet with code from the lecture notes for the course\n",
    "**Permutation Puzzles: A Mathematical Perspective**, by Jamie Mulholland\n",
    "\n",
    "Coures webpage: http://www.sfu.ca/~jtmulhol/math302 <br />\n",
    "Course notes booklet: http://www.sfu.ca/~jtmulhol/math302/notes/302notes.pdf"
   ]
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   "source": [
    "# Lecture 4: Permutations: Cycle Notation\n",
    "\n",
    "<br />\n",
    "\n",
    "## Section 4.7 Working with Permutations in SageMath\n",
    "\n",
    "<br />\n",
    "\n",
    "SageMath uses **disjoint** cycle notation for permutations, and permutation composition occurs left-to-right, which agrees with our convention. There are two ways to write the permutation $\\alpha=(1,3)(2,5,4)$:\n",
    "\n",
    "1. As a text string of disjoint cycles (include quotes): `\"(1,3)(2,5,4)\"` \n",
    "2. As a list of disjoint tuples: `[(1,3), (2,5,4)]`\n"
   ]
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     "text": [
      "(1,4)(2,3) () (2,5,3)\n"
     ]
    }
   ],
   "source": [
    "S5=SymmetricGroup(5)   # symmetric group on 5 objects, and names it S5\n",
    "a=S5(\"(2,3)(1,4)\")     # constructs the permutation (2,3)(1,4) in S5\n",
    "b=S5(\"\")               # constructs the identity permutation in S5\n",
    "c=S5(\"(2,5,3)\")        # constructs the 3-cycle (2,5,3) in S5\n",
    "print a, b, c,"
   ]
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   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1,4)(3,5)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "a*c             # compose permutations by using multiplication sign"
   ]
  },
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   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
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    {
     "data": {
      "text/plain": [
       "(2,3,5)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "c.inverse()     # computes inverse"
   ]
  },
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   "cell_type": "code",
   "execution_count": 4,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 4,
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   "source": [
    "c.order()       # computes order"
   ]
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