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{\bf Math 307 Abstract Algebra \quad Homework 6 \hfill Your name}
\medskip\centerline{
Five points for each question.
}
\begin{enumerate}
\item
(a) Let $H = \langle (1,2,3) \rangle \in A_4$.
Write down all the left cosets of $H$ in $A_4$, and
the right cosets
\quad
of $H$ in $S_4$.
(b) Let $H = \{e^{it}: t \in [0, 2\pi)\} \le \C^*$. Describe geometrically
the left cosets of $H$.
\item Suppose $K$ is a proper subgroup of $H$ and $H$ is a proper subgroup of $G$. If
$|K| = 42$ and $|G| = 420$, what are the possible orders of $H$?
\item Let $G$ be a group with $|G| = pq$, where $p, q$ are primes.
Prove that every proper subgroup of $G$ is cyclic.
Give an example to show that such a group $G$ may not be cyclic.
\item Let $G$ be a group of order $p^2$ for a prime $p$. Show that
$G$ is cyclic or $g^p = e$ for all $g \in G$.
\item Show that a group of order 55 cannot have exactly 20 elements of order 11?
Give a reason for your answer.
[Hint: If $G$ is cyclic, then number of elements of order 11 equal???
If $G$ is not cyclic, then $a \in G$ has order 1, 5, or 11. So, ....]
\item Let $G$ be a group, and $H \le K \le G$. Suppose
$a_1K, \dots, a_r K$ are distinct cosets of $K$ in $G$, and
$b_1H, \dots, b_s H$ are distinct cosets of $H$ in $K$. Prove that
$a_ib_jH$ with $1 \le i \le r, 1 \le j \le s$ are all the distinct
cosets of $H$ in $G$. Deduce that
$$|G:H| = |G:K| \, |K:H|.$$
{\bf Recall} that $H \le G$ is a normal subgroup if $aH = Ha$ for all $a \in G$.
\item (a) Prove that if $H \le G$ and $|G:H| = 2$, then $H$ is normal.
(b) Deduce that if $H \le S_n$ contains a an odd permutation, then
$H$ has a normal subgroup.
\item Let $H \le G$.
(a) Prove that the map $f: aH \rightarrow Ha$ defined by $f(ah) = ha$ is a bijection.
(b) Prove that $H$ is normal if and only if $aHa^{-1} \subseteq H$ for all $a \in G$.
\item (Extra credits) Prove that $A_5$ has no subgroup of order 30.
[Hint: Prove by contradiction. Assume $H \le A_5$ has 30 elements.
Then $A_n-H$ is a left coset as well as a right coset of $H$ in $A_n$.
Argue that $H$ has an element $\sigma_1 = (i_1,i_2)(j_1,j_2)$, and then show that
$\sigma_2 = (i_1,j_2),(i_2,j_1), \sigma_3 = (i_1,j_1)(i_2,j_2) \in H$.
Then argue that $\{\varepsilon, \sigma_1, \sigma_2, \sigma_3\}$ is a 4-element
subgroup of $H$, ...]
\end{enumerate}
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