- Lecture 1: Here is my note for the first lecture.
In this lecture, we emphasized on the fact that groups should be viewed as symmetries of objects;
at the level of set theory the group of symmetries of a set \(X\) is precisely the symmetric group of \(X\),
and it is denoted by \(S_X\); we pointed out how by using rigidity of space one can classify its symmetries. We
worked out the case of the automorphism group of the \(n\)-cycle. We called it the dihedral group,
and it is denoted by \(D_{2n}\); we showed that it has \(n\) rotations and \(n\) reflections. We also classified isometries of the
Euclidean plane based on its rigidity; more precisely we showed that any isometry of the Euclidean plane is a composite of
a translation, a rotation, and/or a reflection. Then we said what a (left) group action of \(G\) on \(X\) is; we write
\(G\curvearrowright X\) if \(G\) acts on \(X\); we pointed out that there is a bijection between the set of left group actions of
\(G\) on \(X\) and \({\rm Hom}(G,S_X)\). Based on this we proved a theorem by Cayley that says any group \(G\) can be embedded into
the symmetric group \(S_G\).
- Lecture 2: Here is my note for the second lecture.
We gave an alternative proof of Cayley's theorem. Let \({\rm Act}(G,X):=\{m:G\times X\rightarrow X|\hspace{1mm} m \text{ is a group action}\}\).
Then we proved that the following functions are inverse of each other:
\[
\Psi:{\rm Act}(G,X)\rightarrow {\rm Hom}(G,S_X),\hspace{1mm} ((\Psi(m))(g))(x):=m(g,x),
\]
and
\[
\Phi: {\rm Hom}(G,S_X) \rightarrow {\rm Act}(G,X),\hspace{1mm} (\Phi(f))(g,x):=(f(g))(x).
\]
Based on this, we mentioned a useful trick: if \(|G|>|X|!\) and \(G \curvearrowright X\), then \(G\) is not simple. Next we mentioned that
if \(G \curvearrowright X\) and \(f:H\rightarrow G\) is a group homomorphism, then we get a group action \(H \curvearrowright X\). Defined
\(G\)-orbit and the stabilizer group \(G_x\) of \(x\in X\). Proved that the set of orbits is a partition of \(X\); denoted this set by \(G\backslash X\) and called the quotient of \(X\) by \(G\). Prove the Orbit-Stabilizer theorem: \(G/G_x\rightarrow G\cdot x, gG_x \mapsto g\cdot x\) is a bijection. Proved that
\(\frac{|X|}{|G|}=\sum_{G\cdot x \in G\backslash X} \frac{1}{|G_x|}\).
- Lecture 3: Here is my note for the third lecture.
Proved that \(G_{g\cdot x}=gG_x g^{-1}\). In particular, if \(x,y\) are in the same \(G\)-orbit, then \(|G_x|=|G_y|\). Proved that \(g\cdot X^h=X^{ghg^{-1}}\); and deduced that \(g\mapsto |X^g|\) is a class function. Proved that the average of fixed points is equal to the size of the quotient space. Proved that if \(G\) acts transitively on \(X\) and \(G\) is finite, then there is \(g\in G\) that has no fixed points. Used this to show if \(G\) is a finite group and \(H\) is a proper subgroup, then \(G\neq \cup_{g\in G} gHg^{-1}\). Pointed out that this is not true for infinite groups. Proved that \(|{\rm Cl}(g)|=[G:C_G(g)]\). Proved the class equation. Defined the normal core of a subgroup, and proved that \([G:{\rm core}(H)]|[G:H]!\). Used this result to show that if \([G:H]=p\) is the smallest prime factor of \(G\), then \(H\) is a normal subgroup of \(G\). Defined the normalizer of a subgroup and proved that the number of conjugates of a subgroup \(H\) is equal to \([G:N_G(H)]\).
- Lecture 4: Here is my note for the fourth lecture.
We proved (the main theorem of actions of finite \(p\)-groups) if \(|G|=p^n\), \(G\curvearrowright X\), then \(|X|\equiv |X^G| \pmod{p}\). This was used in several ways. We proved, if \(|H|=p^m\) and \(p| |G/H|\), then \(p| |N_G(H)/H|\). If \(|G|=p^n\) and \(1\neq N\unlhd G\), then \(Z(G)\cap N\neq 1\); in particular, \(Z(G)\neq 1\). Cauchy's theorem was proved next: if \(p| |G|\), then there is \(g\in G\) that has order \(p\). We deduced that order of a finite \(p\)-group is \(p^n\) for some \(n\). Then the first Sylow theorem was proved: if \(p^m| |G|\), then there are \(P_1\unlhd P_2\unlhd \cdots \unlhd P_m \leq G\) such that \(|P_i|=p^i\) for \(1\le i\le m\). Then we defined Sylow \(p\)-subgroups and \({\rm Syl}_p(G)\), and proved the second Sylow theorem: if \(P_0\in {\rm Syl}_p(G)\), \( Q\leq G\), and \(|Q|=p^m\), then there is \(g\in G\) such that \(gQg^{-1} \subseteq P_0\); in particular, \(G\curvearrowright {\rm Syl}_p(G)\) transitively via conjugation. We deduced that if \(P\in {\rm Syl}_p(G)\), then \(|{\rm Syl}_p(G)|=[G:N_G(P)]\). Using the second Sylow theorem, we showed
\({\rm Syl}_p(N_G(P))=\{P\}\) if \(P\in {\rm Syl}_p(G)\). And then proved that \(N_G(N_G(P))=N_G(P)\) if \(P\in {\rm Syl}_p(G)\). Next time we will prove the third Sylow theorem.
- Lecture 5: Here is my note for the fifth lecture.
We proved the 3rd Sylow theorem. Using Sylow theorems, we described the possible group structures of a group of order \(pq\) where \(p< q\) are primes; in
particular, we showed that if \(p\nmid q-1\), then such a group is cyclic. Along the way we showed that \(|HK|=\frac{|H||K|}{|H\cap K|}\) for two finite subgroups \(H\) and \(K\). Next we showed a group of order \(p(p-1)\) has a normal subgroup of order \(p\). Finally we showed that a group of order \(p(p+1)\) has a normal subgroup of order either \(p\) or \(p+1\).
- Lecture 6: Here is my note for the sixth lecture.
We learned about exact sequences, short exact sequences (SES), split SES, a criterion for a SES to split, semi-direct product and its connection with split SESs.
- Lecture 7: Here is my note for the seventh lecture.
We saw how to use split SES and semi-direct product to describe groups of order \(pq\) where \(p< q\) are prime. We also recalled that \({\rm Aut}(\mathbb{Z}/n\mathbb{Z})\simeq (\mathbb{Z}/n\mathbb{Z})^{\times}\) and outlined its proof. Next we stated Schur-Zassenhaus theorem (a SES \(1\rightarrow N \rightarrow G \rightarrow H\rightarrow 1\) splits if \(\gcd(|N|,|H|)=1\)); and showed it is enough to prove Schur-Zassenhaus when \(N\) is abelian. After this, we started analyzing the symmetric group \(S_n\).
- Lecture 8: Here is my note for the eighth lecture.
We proved the existence and uniqueness of the cycle decomposition of a permutation. Defined cycle type and showed that conjugacy classes are uniquely determined by cycle types. Showed linking equation; defined transposition; and defined the sign function.
- Lecture 9: Here is my note for the ninth lecture.
Proved that the sign function is a group homomorphism. Mentioned the connection with crossing number, and word length with respect to certain generating set of the symmetric group. Defined odd and even permutations and mentioned its connection with the 15-puzzle. Proved that \(A_n\) is simple if \(n\ge 5\).
- Lecture 10: Here is my note for the tenth lecture.
Proved that a group of order \(2m\) has a characteristic subgroup of order \(m\) if \(m\) is odd. Defined composition series and composition factors of a
group and proved Jordan-Holder theorem for finite groups. Defined derived series and showed that the \(k\)-th group in the derived series is trivial if and only if there is a chain of normal subgroups that has length \(k\) and all the successive quotients are abelian (defined solvable).
- Lecture 11: Here is my note for the eleventh lecture.
Proved that a finite group is solvable if and only if all of its composition factors are cyclic groups of prime order. Defined lower and upper central series. Proved that \(\gamma_{1+c}(G)=1\) if and only if \(Z_c(G)=G\); and called such a group nilpotent. Proved that if \(H\) is a proper subgroup of a nilpotent group \(G\), then \(H\) is a proper subgroup of \(N_G(H)\). Proved that if \(G\) is a finite group, then the following are equivalent: (1) \(G\) is nilpotent, (2) for any \(p| |G|\), \(G\) has a unique Sylow \(p\)-subgroup, (3) \(G\simeq \prod_{i=1}^m P_i\) where \(P_i\) is a finite \(p_i\)-group and \(p_i\)'s are distinct primes, (4) Any maximal subgroup of \(G\) is normal.
- Lecture 12: Here is my note for the twelfth lecture.
Proved some further properties of nilpotent groups, e.g. if \(G\) is nilpotent and \(N\) is a non-trivial normal subgroup of \(G\), then \(Z(G)\cap N\) is non-trivial. Then Frattini subgroup of a group was defined. It is observed that Frattini subgroup is a characteristic subgroup; if \(f:G\rightarrow H\) is an onto group homomorphism, then \(f(\Phi(G))\subseteq \Phi(H)\); if \(V\) is an abelian group and order of any non-trivial element is a fixed prime number \(p\), then \(V\) is a vector space over \(\mathbb{Z}/p\mathbb{Z}\), and \(\Phi(V)=0\). Next we proved that, if \(G\) is a finite \(p\)-group, then \(\Phi(G)=G^p[G,G]\). Then we defined the free monoid and proved its universal property. Defined the free product of a family of groups.
- Lecture 13: Here is my note for the thirteenth lecture.
Proved the universal property of free product of groups. Defined free group, and proved its universal property.
- Lecture 14: Here is my note for the fourteenth lecture.
We started with Ping-Pong Lemma: Suppose \(G\) is a group, \(G_1,G_2\) are two subgroups of \(G\), and \(|G_1|\ge 3, |G_2|\ge 2\). Suppose
\(G\curvearrowright X\), \(X_1,X_2\subseteq X\), \(X_1\not\subseteq X_2\), and \(X_2\not\subseteq X_1\). Suppose
\((G_1\setminus \{1\})\cdot X_2\subseteq X_1\) and \((G_2\setminus \{1\})\cdot X_1\subseteq X_2\). Then \(\langle G_1\cup G_2\rangle\simeq G_1\ast G_2\).
; and then used it proved the following results: \(\left[\begin{smallmatrix}1&2\\0&1\end{smallmatrix}\right]\) and \(\left[\begin{smallmatrix}1&0\\2&1\end{smallmatrix}\right]\) freely generate a subgroup of
\({\rm SL}_2(\mathbb{Z})\). proved that the group generated by \(\overline{\left[\begin{smallmatrix}1&2\\0&1\end{smallmatrix}\right]}\) and
\(\overline{\left[\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right]}\) is isomorphic to \(\mathbb{Z}\ast \mathbb{Z}/2\mathbb{Z}\), where
\(\overline{g}:=gZ({\rm SL}_2(\mathbb{R}))\in {\rm PSL}_2(\mathbb{R})\); we proved that if
\(a=\left[\begin{smallmatrix}\lambda&0\\0&1/\lambda\end{smallmatrix}\right]\) where \(\lambda>1\) and
\(b\in {\rm SL}_2(\mathbb{R})\) is not upper or lower triangular matrix and it is not of the form
\(\left[\begin{smallmatrix}0&x\\-1/x&0\end{smallmatrix}\right]\), then, for large enough \(n\),
\(a^n\) and \(ba^nb^{-1}\) freely generate a subgroup, and so \(\langle a,b \rangle\) has a non-commutative free subgroup. Tits alternative was also mentioned. At the end of the lecture, presentation of a group was defined.
- Lecture 15: Here is my note for the fifteenth lecture.
We discussed what presentation of a group is; found presentation of the dihedral group \(D_{2n}\). Then we started ring theory with a historical note. Then we defined the matrix ring with entries in a given ring \(R\); the monoid ring \(RM\) of a monoid \(M\) with coefficients in a ring \(R\); we also viewed elements of this ring as functions and interpreted its ring multiplication as the convolution of functions. Next we defined ring of polynomials with indeterminants \(x_1,\ldots,x_n\) and viewed it as the monoid ring \(R(\mathbb{Z}^{\ge 0})^n\).
- Lecture 16: Here is my note for the sixteenth lecture.
We recalled the factor ring \(R/I\) and the first isomorphism theorem. Viewed polynomials as functions and defined the evaluation homomorphisms. Recalled what it means to say an ideal is generated by set; specially when an ideal is finitely generated. Recalled zero-divisor, \(a|b\), group of units, integral domain, and field. Observed that any field is an integral domain. Proved a finite ring is a field if and only if it is an integral domain. Recalled that if \(D\) is an integral domain, then \(\deg fg=\deg f+\deg g\); and deduced that: \(D[x]^{\times}=D^{\times}\), and \(D\) is an integral domain if and only if \(D[x]\) is an integral domain. We recalled the defintion of prime and maximal ideals.
- Lecture 17: Here is my note for the seventeenth lecture.
Proved that in ideal \(I\) is prime if and only if \(A/I\) is an integral domain; and ideal \(I\) is maximal if and only if \(A/I\) is a field. Deduced that \({\rm Max}(A)\subseteq {\rm Spec}(A)\) and for an ideal \(I\) of finite index we have \(I\in {\rm Spec}(A)\) if and only if \(I\in {\rm Max}(A)\). Next in order to show existence of ``lots" of primes, we said what Zorn's lemma is: mentioned what a partially ordered set, chain, upper bound and maximal elements are. Then we proved: suppose \(S\) is a multiplicatively closed subset of \(A\), \(\mathfrak{a}\unlhd A\), and \(S\cap \mathfrak{a}=\varnothing\). Let \(\Sigma_{\mathfrak{a},S}:=\{\mathfrak{b} \unlhd A| \mathfrak{b}\cap S=\varnothing, \mathfrak{a}\subseteq \mathfrak{b}\}\). Then (1) \(\Sigma_{\mathfrak{a},S}\) has a maximal element, (2) any maximal element of \(\Sigma_{\mathfrak{a},S}\) is a prime ideal. In particular \(\Sigma_{\mathfrak{a},S}\cap {\rm Spec}(A)\neq \varnothing\). Using this we proved for any proper ideal \(\mathfrak{a}\) there is a maximal ideal \(\mathfrak{m}\) which contains \(\mathfrak{a}\) as a subset. Next we defined the nilradical \({\rm Nil}(A)\) and proved that \({\rm Nil}(A)=\cap_{\mathfrak{p}\in {\rm Spec}(A)} \mathfrak{p}\).
- Lecture 18: Here is my note for the eighteenth lecture.
We defined Euclidean Domain. Observed that \(\mathbb{Z}\) and \(F[x]\) when \(F\) is a field are EDs. Next we proved that \(\mathbb{Z}[i]\) is a ED. Then we proved a ED is a PID. To do ``number theory" with rings, we defined irreducible and prime elements, and being associates. Then we proved in an integral domain \(D\), (1) \(a\in D\) is irreducible if and only if \(a\neq 0\) and \(\langle a\rangle\) is maximal among proper principal ideals, (2) \(a\in D\) is prime if and only if \(a\neq 0\) and \(\langle a\rangle\) is prime, (3) \(a\) and \(b\) are associates if and only if \(\langle a\rangle=\langle b\rangle\). Then we showed in an integral domain prime implies irreducible; and in a PID \(a\) is prime if and only if \(a\) is irreducible. Noetherian rings were defined, and it was proved that a ring is Noetherian if and only if it satisfies the ascending chain condition.
- Lecture 19: Here is my note for the nineteenth lecture.
We proved that a ring is Noetherian if and only if any of its ideals is finitely generated. Next we proved in a Noetherian integral domain and non-zero non-unit element can be written as a product of irreducibles. Now assuming in an integral domain \(D\) any non-zero non-unit element can be written as a product of irreducibles, we proved that \(D\) is a UFD if and only if any irreducible in \(D\) is prime. And so for a Noetherian integral domain \(D\), \(D\) is a UFD if and only if any irreducible in \(D\) is prime. In particular, any PID is a UFD. Next we seek properties for a ring that can be passed on to the ring of polynomials, and showed that being a PID is not one of such properties; in fact we showed \(A[x]\) is a PID if and only if \(A\) is a field. To show this result, first we proved that, if \(D\) is a PID, then \({\rm Spec}(D)=\{0\}\cup {\rm Max}(D)\). And \(A[x]/\mathfrak{a}[x]\simeq (A/\mathfrak{a})[x]\). At the end, we started proof of Hilbert's basis theorem: \(A\) is Noetherian implies that \(A[x]\) is Noetherian; and presented its informal proof.
- Lecture 20: Here is my note for the twentieth lecture.
We finished proof of Hilbert's basis theorem; deduced that any finitely generated \(k\)-algebra is Noetherian if \(k\) is a field. Then defined \(p\)-valuations and g.c.d. for a UFD, and mentioned their basic properties. Defined the content of a polynomial and a primitive polynomial in \(D[x]\) where \(D\) is a UFD. Then proved Gauss's lemma: \(c(fg)=c(f)c(g)\). Using Gauss's lemma we proved: suppose \(D\) is a UFD and \(F\) is its field of fractions. Suppose \(f(x)\in D[x]\) has degree at least 1 and \(f(x)=\prod_{i=1}^n f_i(x)\) for some \(f_i(x)\in F[x]\). Then there are \(c_i\in F\) such that \(\prod_{i=1}^n c_i=1\) and \(c_i f_i(x)\in D[x]\). In particular, if \(f(x)\in D[x]\) is reducible in \(F[x]\), then it is reducible in \(D[x]\). In the next quarter this will be used to show \(D\) is a UFD if and only if \(D[x]\) is a UFD.
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