Math 3110 Fall 2011 -- Homework Problem Set #3

Due: Wednesday, October 12^th

Chapters 4 & 5

1. Chapter 4 #14: Suppose that a cyclic group \( G \) has exactly 3 subgroups: \( G \) itself, the trivial subgroup \( \{ e \} \), and a subgroup of order 7.
   • What is \( |G| \)?
   • What can you say if 7 is replaced with \( p \) where \( p \) is prime?
   
   
2. Chapter 4: #54: Let \( a, b \in G \) (a group) and suppose \( |a| \) and \( |b| \) are relatively prime. Show that \( \langle a \rangle \cap \langle b \rangle = \{ e \} \).
   
   
3. Chapters 1-4 Supplementary Problems #6: Let \( a,b \in G \) where \( G \) is a finite group.
   • Give a concrete example which shows that \( |ab| \not= |a| \cdot |b| \).
   • Prove that \( |ab|=|ba| \).
   
   
   
4. For each of the following permutations: write the permutation as a product of disjoint cycles, find its inverse, find its order, write it as a product of transpositions, and state whether it is even or odd.
   1. \( (1432)(56)(254) \)
      
      
   2. \( (1234)(1423)(246) \)
      
      
   3. \( (12)(345)(1357) \)
   
   
   
5. Let \( \sigma = (a_1\,a_2\,\cdots a_\ell) \in S_n \) and \( \tau \in S_n \). Prove that \( \tau \sigma \tau^{-1} = ( \tau(a_1)\,\tau(a_2)\,\cdots\,\tau(a_\ell)) \).
   [This quickly implies that any permutation and its conjugates share the same cycle structure.]
   
   Example: \( \sigma = (1423) \) and \( \tau = (15)(463) \) then \( \tau \sigma \tau^{-1} = (\tau(1)\,\tau(4)\,\tau(2)\,\tau(3)) = (5624) = (2456) \).
   
   
6. Does \( S_7 \) have an element of order 10? If so, find one. If not, explain why not.
   Does \( S_7 \) have an element of order 9? If so, find one. If not, explain why not.