Problem Set 1, Due Friday Jan 24th in Class

Appendix A: 12, 23; Appendix B: 9; Appendix C: 3, 6.

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 $i = 1, \dots, s$.

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Problem Set 2, Due Friday Jan 31rd in Class

Chap 2: A2,A3, B2, B6 ; Chap 3: A3, C1, C2 C3; Chap 4: A4, B2, H1.

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Problem Set 3, Due Friday Feb 7th in Class

Chap 5: A2, A4, A6, C6, D1, E1, Chap 6: A2, A5, C4, D5, E6, G2; Chap 7: B1, F2.

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 $gcd(m , n)=1$.

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Problem Set 4, Due Friday Feb 14th in Class

Chap 8: A2(d), A3(d), B6, C2, G3; Chap 10:A3, B3, C6, H3. Chap14: A4, D4.

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Problem Set 5, Due Monday Feb 24th in Class

Chap 9: B3, C4, H3 ; Chap 11: C1, E2. Chap 12: B10, D3, and the following questions:

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 $Aut (G)$.

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Problem Set 6, Due Friday March 6th in Class

Chap 13: A3, B3, C4, D1; Chap 15: A2 (see page 71 for $S_3$ and its elements), C2, E1. Chap 16: A3, E2, E3.

Extra Credits: Chap 16: H7.

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Problem Set 7, Due Friday March 13th in Class

Chap 16: I 1, I2, I4, I5. and the following questions:

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.

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Problem Set 8, Due Friday March 27th in Class

Chap 17: A2, A6. G1, I5; Chap 18: A3, C6, D4, D5, F7, G2.

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 $A$.