0. Given integers $m$ and $n$, show that $gcd(m,n)$ is the
smallest positive integer of the form $am+bn$, with integers $a$
and $b$.
1. Let $S$ be a square lying flat on a table. Let $F$ be the
collection of its four edges. List all the bijections on $F$ which
arise from rotations of the square, lying flat on the table. Do
the same, with reflections allowed as well.
Verify that these bijections form a subgroup of cardinality 4
(respectively 8) of the group of all 24 bijections of the four
edges.
2. Let $f:Z_m \rightarrow Z_n $ be the map $f(i)=i mod n$. Check
that the map is well-defined only when $n$ divides $m$. And in
this case, $f$ is a group homomorphism.
3. An element $g\in G$ is said to generate the group if every
element is obtained from $g$ as some suitable power of it.
Demonstrate a group which cannot be generated by a single element.
Show that $Z_m $ is generated by a single element. What are the
possible generators of $Z_6 $? And of $Z_7 $. And of $Z_n $, in
general?
4. Let $S\subseteq G$ be an arbitrary subset of a group $G$. Let
$Z(S)=\{ g\in G| gs=sg \:for \: all \: s\in S \}$. Show that
$Z(S)$ is a subgroup of $G$.