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\pagestyle{myheadings} \markright{Math 3340 (Spring 2021) \hspace{2in}
(HW3)}

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  {\large Homework \# 3 $\quad$ Math 3340 Spring 2021

    \medskip
    
    Due at the end of the day 4 Mar 2021, at midnight.
  }

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Please submit your completed homework via gradescope on canvas.
I encourage you to work with your classmates on this homework (except for the journal entries!)
When you submit your work, please {\bf list your collaborators}.
(Your grade will not be affected.)
Even if you work in a group, you should write up 
your solutions {\bf yourself}! 
You should include all computational details, and proofs should be carefully 
written with full details.
As always, please write {\bf neatly and legibly} (feel free to use \LaTeX\ to write up your solutions, if you wish!).

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%%   \noindent {\bf Reading.} {\em The Secret to Raising Smart Kids} by Carol Dweck, in Scientific American.  You can find it
%% by Google, or in the files section on our canvas site (it is also linked to from the syllabus).

%% \vskip 0.1in

\noindent {\bf Journal entry.} There is no journal entry this week.

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\noindent {\bf Exercises.} 

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\begin{enumerate}

\item In this course, we accept (as an axiom) the well-ordering principle, stated below.  We let $\N$
  denote the set of all non-negative integers (natural numbers.  But note, zero is an element of this set!).

  {\bf Well-ordering principle}: Every non-empty subset of $\N$ contains a minimal element.

  In this problem, we assume this axiom (as we will for the entire
  course).  Using this, we will prove that the method of proof by mathematical induction holds (i.e. the 2 following statements),
  and then apply it in an example.
  
  \medskip
      {\bf The principle of mathematical induction I}: Let $P_1, P_2,  \ldots, $ be a sequence of propositions
      (i.e. each $P_k$ is either true, or false).
      Suppose that (1) $P_1$ is true, and (2) For all positive integers, if $P_k$ is true, then $P_{k+1}$ is true.
      Then $P_m$ is true for all positive integers $m$.

  {\bf The principle of mathematical induction II}: Let $P_1, P_2,  \ldots, $ be a sequence of propositions.
  Suppose that (1) $P_1$ is true, and (2) For all positive integers $k$, if $P_m$ is true for all $m \le k$, then $P_{k+1}$ is true.
  Then $P_m$ is true for all positive integers $m$.
  
  \babc
\item Prove the first principle of mathematical induction.  You might want to start with
  ``Let $T = \{ n \in \N \mid n \ge 1 \ \mbox{and}\  P_n\ \mbox{is not true}\}$.''
\item Prove the second principle of mathematical induction.  Once again, you might want to start with
  ``Let $T = \{ n \in \N \mid n \ge 1 \ \mbox{and}\ P_n\ \mbox{is not true}\}$.''
\item Prove using induction: For all positive integers $n$,
  \[ 1^2 + 2^2 + \ldots + n^2 = \frac{n(n+1)(2n+1)}{6}. \]
  (The proposition $P_n$ is that this formula holds for the given $n$).
  \eabc

  
\item
  \babc
  \item 
    Let $G$ be a group, and suppose $a, b \in G$ commute ($ab = ba$).  Show that
    if $o(a)$ and $o(b)$ are finite and relatively prime, then $o(ab) = o(a) o(b)$.
  \item Let $\sigma \in S_n$ be a permutation, of cycle type $(n_1, n_2, \ldots, n_r)$,
    where $n_1 \ge n_2 \ge \ldots \ge n_r \ge 2$.
    (this means that the unique representation of $\sigma$ as a product of disjoint cycles
    consists of cycles
    of length $n_1, n_2, \ldots, n_r$).  For example $\sigma = (1,2,3)(4,5)(6,7,8)(9,10,11,12)$
    has cycle type $(4,3,3,2)$.  Show that the order of $\sigma$ is the least common multiple of
    $\{ n_1, \ldots, n_r\}$.
  \eabc
  
\item Let $G$ be a group of order 6. We know from HW2 that $G$ must contain an odd number of elements of
  order 2 (i.e., 1, 3, or 5).
  \babc
\item Show that this number cannot be 5 (if it is, show that $G$ would contain a subgroup of order 4).
\item If $G$ is not cyclic, show that there exists an element $a \in G$ of order 2, and $b \in G$
  of order 3, such that $ab \ne ba$.
\item If $G$ is not cyclic, show that the multiplication table of $S_3$ and of $G$ are the ``same''
  (that is, one can name all the elements of each group, so that the multiplication tables are the same)
  \eabc
  

\item Let $G$ be a group, and let $S \subseteq G$ be a subset.  In this problem, we define and
  consider the subgroup $\langle S \rangle$ of $G$ generated by $S$.
  \babc
\item Show that the intersection of a (possibly infinite) collection of subgroups of $G$ is also a subgroup.
\item 
  Define
  \[ \langle S \rangle := \bigcap_{\{ H \le G \mid S \subseteq H \}} H \]
  to be the intersection of all subgroups of $G$ containing $S$.
  Deduce that there is a unique minimal subgroup containing $S$.
  This is called the subgroup
  of $G$ {\bf generated by} $S$ (or, the subgroup that $S$ generates).
\item A word in $S$ is an element of the form $g = w_1 w_2 \ldots w_r$, for any $r \ge 0$, where each
  $w_i$ is in $S$, or $w_i^{-1} \in S$ (or both), and if $r = 0$, we mean $g = e$.
  Show that the group generated by $S$
  is precisely the set of words in $S$.
  
\item Show, using induction, that $S = \{ (1,2), (2,3), \ldots, (n-1,n) \}$ generates $S_n$,
  for all $n \ge 2$.
  \eabc
  
\item In each of the following cases, determine if $H$ is a subgroup of $G$ (and prove your assertions).
  \babc

  \item Let $G \le S_n$ be a subgroup.
  \[ H = \{ \sigma \in G \mid \sigma(1) = 1 \}. \]

\item
  Fix a positive integer $n > 2$. Consider the group $G = S_n \times S_n$.
  \[ H = \{ (\sigma, \tau) \in G \mid \sigma(1) = \tau(1) \}. \]

\item Let $G$ be a group, let
  \[ H = Z(G) = \{ h \in G \mid gh = hg, \mbox{ for all $g \in G$} \} \]
  (called the center of $G$).

\item Let $G$ be a group, and let $a \in G$. Let
  \[ H = C_G(a) = \{ g \in G \mid ga = ag \}\]
  ($H$ is called the centralizer of $a$ in $G$).
  \eabc


  
\end{enumerate}

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