Prove the $uniqueness$ of prime decompostion: Let $n \in \mathbb Z$ and $n > 1$. Suppose that there are two prime decomposition of $n$: \[ n = p _1^{r_1} \cdots p_s^{r_s} = q_1^{l_1} \cdots q_t^{l_t} \] Show that after reordering $p_i$ and $q_j$, we have $s=t$, $p_i = q_i$ and $r_i = l_i$, for all $i = 1, \dots, s$.
Extra Credits: Let $m , n \in \mathbb Z$, then $\mathbb Z = \langle m , n \rangle $ if and only of $m$ and $n$ are relatively prime. That is $gcd(m , n)=1$.
1) Let $\pi: = (a_1 a_2 \cdots a_t)$ be a cycle inside $S_n$. Find the order of $\pi$.
2) Let $A$ be the set of all groups. Define $G\sim H$ if $G$ is isomorphic to $H$. Show that $\sim$ is an equivalence relation on $A$.
Extra Credits: Let $X$ be a set. Set $Aut(X): = \{f: X \to X | f \text{ is bijective}\}$. Prove that
a) $Aut (X)$ is a group with group law $f g = f\circ g$.
b) Let $G$ be a group. Then $G$ is isomorphic to a subgroup of $Aut (G)$.
Extra Credits: Chap 16: H7.
1) Let $f: G \to H$ be an onto homomorrphism with $K = \text{Ker} (f)$. We know there is a bijection between the following two sets:
{subrgroup $L \subset G$ such that $K \subset L$} $\longrightarrow$ { subgroup $N \subset H$} via $ L \mapsto f(L)$.
Show that $L$ is a normal subgroup $L \subset G$ if only if $f(L) \subset H$ is normal.
2) Show that $M \times N \simeq G$ if and only if there exists two normal subgroups $H_1 , H_2 \subset G$ so that
a) for all $x \in H_1$ and $y \in H_2$, $xy = yx$.
b) $H_1 \cap H_2 = \{e\}$
c) for any $z \in G$, there exists $x\in H_1$, $y \in H_2$ so that $z= xy$.
In fact, if $f: M \times N \simeq G$ then $H_1 = f (M \times \{e_2\})$ and $H_2 = f(\{e_1\} \times N)$.
Extra Credits: Chap 16: O1--O4.
Extra Credits: Let $A=M_{n \times n}(\mathbb R)$ be the ring of all $n \times n$ matrices with real entries. Show that only ideals of $A$ are trivial ones: $\{0\}$ and $A$.