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{\bf Math 307 Abstract Algebra \quad Homework 5 \hfill
Your name}
\bigskip
Five points for each question.
\begin{enumerate}
\item Let $(\R^+, \cdot)$ be the group of positive real number under multiplication.
Show that $\phi: \R^+\rightarrow \R^+$ defined by $\phi(x) = \sqrt{x}$
is an group isomorphism.
\item Show that the following pair of groups are not isomorphic.
(a) $(\Q,+), (\R,+)$. \quad (b) $(\R^+, \cdot), (\R^*, \cdot)$. \quad
(c) $(\R^*,\cdot), (\C^*,\cdot)$.
[Recall that $\R^*$ and $\C^*$ are the sets of nonzero real numbers and nonzero complex numbers.]
\item Show that $G = \{e^{it}: t\in [0, 2\pi)\}$ under multiplication contains subgroups isomorphic
to $(\Z,+)$ and $(\Z_n,+)$ for any $n \in \N$, and show that $G$ is not a cyclic group.
\item Suppose $\phi_1, \phi_2$ are automorphisms of a group $G$. Show that
$H = \{g \in G: \phi_1(g) = \phi_2(g)\}$ is a subgroup of $G$.
\item Let $G$ be a group and $a \in G$.
Suppose $|a| = n$ and
the inner automorphism $\phi_a: G \rightarrow G$ defined by
$\phi_a(x) = axa^{-1}$ has order $m$ in ${\rm Aut}(G)$.
Show that $m | n$.
Give an example of $G$ and $a$ so that $1 < m < n$.
[Hint: Show that $(\phi_a)^n$ is the identity map, and ....]
\item Suppose $G$ is a group of order $n$, and $k \in \N$ is relatively prime to $n$.
Show that $g: G\rightarrow G$ defined by $g(x) = x^k$ is one-one. If $G$ is Abelian,
show that $g$ is an automorphism.
[Hint: To prove that $g$ is bijective, it
suffices to show that $g$ is 1-1. It suffices to show that $g(x) = e$ if and only if $x = e$.]
\item Suppose $1 \le i < j\le n$.
(a) Show that $(i,j) = (j,j-1)(j-1,j-2) \cdots (i+1,i) (i+1,i+2) \cdots (j-1,j)$.
(b) Show that every element $\sigma \in S_n$ is a product of transpositions of the form
\medskip\hskip 1.5in
$(1,2), (2,3), \dots, (n-1,n)$.
[Hint: To prove (a),
show that the bijections on right sides will exchange $i$ and $j$, and fixes
all other $k$.]
\item (Extra 5 points)
(a) Show that every $\sigma \in S_n$ is a product of
the $n$-cycle $\alpha = (1,2, \dots, n)$ and
$\tau = (1,2)$.
[Hint: It suffices to use $\alpha, \tau$ to generate all transpositions of the form
$(i,i+1)$
\ \ for $i = 1, \dots, n-1$.
To this end, show that for $k = 1, \dots, n-1$,
$\alpha^k \tau \alpha^{-k} = (k,k+1)$, equivalently, $\alpha^k \tau = (k,k+1)\alpha^k$]
(b) (Open problem) In (a), determine the minimum number of $\alpha$ and $\tau$
needed for a given $\sigma$.
(c) (Conjecture) For $n > 3$, every $\sigma$ is a product of no more than ${n\choose 2}$
permutations in the set
\qquad $\{\alpha, \alpha^{-1}, \tau\}$.
\end{enumerate}
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