#!/usr/bin/env python
# coding: utf-8

# This is an ipython (jupyter) worksheet with code from the lecture notes for the course
# **Permutation Puzzles: A Mathematical Perspective**, by Jamie Mulholland
# 
# Coures webpage: http://www.sfu.ca/~jtmulhol/math302 <br />
# Course notes booklet: http://www.sfu.ca/~jtmulhol/math302/notes/302notes.pdf

# # Lecture 4: Permutations: Cycle Notation
# 
# <br />
# 
# ## Section 4.7 Working with Permutations in SageMath
# 
# <br />
# 
# 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 $\alpha=(1,3)(2,5,4)$:
# 
# 1. As a text string of disjoint cycles (include quotes): `"(1,3)(2,5,4)"` 
# 2. As a list of disjoint tuples: `[(1,3), (2,5,4)]`
# 

# In[1]:


S5=SymmetricGroup(5)   # symmetric group on 5 objects, and names it S5
a=S5("(2,3)(1,4)")     # constructs the permutation (2,3)(1,4) in S5
b=S5("")               # constructs the identity permutation in S5
c=S5("(2,5,3)")        # constructs the 3-cycle (2,5,3) in S5
print a, b, c,


# In[2]:


a*c             # compose permutations by using multiplication sign


# In[3]:


c.inverse()     # computes inverse


# In[4]:


c.order()       # computes order


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