- 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 the rigidity of the Euclidean plane
one can show any symmetry of the Euclidean plane is a composite of a translation, a rotation, and/or
a reflection; we mentioned the automorphism group of a graph as yet another example of a group of
symmetries of an object; the automorphism group of the \(n\)-cycle is defined to be the dihedral group,
and it is denoted by \(D_{2n}\); we did some computation in \(D_{10}\), and pointed out certain
connection between fixed point(s) of a conjugate \(\tau\sigma\tau^{-1}\) of an element \(\sigma\) and
the fixed point(s) of \(\sigma\); 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 \({\rm Hom}(G,S_X)\).
- Lecture 2: Here is my note for the second lecture.
In this lecture, we learned about various examples of group actions: for any group \(G\),
\(G\curvearrowright G\) by the left translations ; (The induced group action)
suppose \(G\curvearrowright X\) and \(H\xrightarrow{f} G\) is a group homomorphism; then the
following defines a group action of \(H\) on \(X\), \(h\ast x:=f(h)\cdot x\);
\(G\curvearrowright G\) by conjugations; that means \(g\cdot g':=gg'g^{-1}\);
\({\rm Aut}(G)\curvearrowright G\); \({\rm Aut}(G)\curvearrowright \{H| \hspace{1mm} H\leq G\}\);
\({\rm Aut}(G)\curvearrowright \{H|\hspace{1mm} H\leq G, [G:H]=n\}\) if non-empty;
\(c:G\rightarrow {\rm Aut}(G), c(g):=c_g\) where \(c_g(g'):=gg'g^{-1}\) is a group homomorphism,
and so we get induced group actions of \(G\) on the above (sub)sets of subgroups of \(G\).
Then we used the action of \(S_X\) on \(X\) to attach a group action of \(G\) on \(X\) to a given
\(f\in {\rm Hom}(G,S_X)\); and pointed out at this is a bijection, and wrote its inverse as well:
let \(A_{G,X}:=\{m:G\times X\rightarrow X|\hspace{1mm} m \text{ is a group action}\}\). Then the following
functions are bijective:
\[
\Psi:A_{G,X}\rightarrow {\rm Hom}(G,S_X),\hspace{1mm} ((\Psi(m))(g))(x):=m(g,x),
\]
and
\[
\Phi: {\rm Hom}(G,S_X) \rightarrow A_{G,X},\hspace{1mm} (\Phi(f))(g,x):=(f(g))(x).
\]
- Lecture 3: Here is my note for the third lecture.
In this lecture, we finally saw an outline of the bijection between the set of group actions
of \(G\) on \(X\) and \({\rm Hom}(G,S_X)\); then we deduced the Cayley's theorem; the \(G\)-
orbit of \(x\) is defined; assuming that \(G\curvearrowright X\) we proved that the stabilizer
\(G_x\) of \(x\in X\) is a subgroup of \(G\), where \(G_x:=\{g\in G|\hspace{1mm} g\cdot x=x\}\); then we
proved the following is a bijection \(G/G_x\rightarrow G\cdot x, gG_x\mapsto g\cdot x\); we proved
that the set \(_G\!\backslash\!\!^X:=\{G\cdot x|\hspace{1mm} x\in X\}\) of \(G\)-orbits is a partition of \(X\).
- Lecture 4: Here is my note for the forth lecture.
In this lecture, we defined a free group action, and proved that, if \(G\curvearrowright X\) freely and
\(X\) is a finite set, then \(|_G\!\backslash\!\!^X|=|X|/|G|\); as a corollary we deduced the Lagrange theorem;
we showed that if \(G\) and \(X\) are finite and \(G\curvearrowright X\), then
\(|X|/|G|=\sum_{[x]\in _G\!\backslash\!\!^X} 1/|G_x|\); for a group action
\(G\curvearrowright X\) and \(g\in G\), we denote the set of fixed points of \(g\) by \(X^g\); then we proved
the Burnside theorem which states that, if \(G\) and \(X\) are finite, then the average of the number of fixed
points of elements of \(G\) is the number of elements if the quotient space \(_G\!\backslash\!\!^X\); that means
\[
|_G\!\backslash\!\!^X|=\frac{1}{|G|}\sum_{g\in G}|X^g|;
\]
we defined a transitive group action, and using the Burnside theorem we proved that, if
\(G\) and \(X\) are finite and \(G\curvearrowright X\) transitively, then there is \(g\in G\)
such that \(X^g\) is empty; as a corollary of this result, we deduced, if \(G\) is a finite group and
\(H\) is a proper subgroup, then \(\bigcup_{g\in G}gHg^{-1}\neq G\); it was explained that this statement fails
for infinite groups.
- Lecture 5: Here is my note for the fifth lecture.
In this lecture, for a group action \(G\curvearrowright X\) we let \(X^G:=\{x\in X|\hspace{1mm} \forall g\in G, g\cdot x=x\}\);
so \(X^G=\cap_{g\in G}X^g\); we observed that, if \(X\) is a finite set, then
\(|X|=|X^G|+\sum_{G\cdot x\in _G\!\backslash\!\!^X, x\not\in X^G} [G:G_x]\); for group action
\(G\curvearrowright G\) by conjugations, we observed that the orbit of \(g\) is the
conjugacy class \({\rm Cl}(g)\) of \(g\) in \(G\); the stabilizer group of \(g\) is
the centralizer \(C_G(g):=\{g'\in G|\hspace{1mm} g'g=gg'\}\) of \(g\) in \(G\); the set
\(G^G\) is the center \(Z(G)\); so \(|{\rm Cl}(g)|=[G:C_G(g)]\), and if \(G\) is a finite group, we get
\[
|G|=|Z(G)|+\sum_{|{\rm Cl}(g)|>1} [G:C_G(g)].
\]
(this is called the class equation); For the group action \(G\curvearrowright \{H|\hspace{1mm} H\leq G\}\)
by conjugations, the orbit of \(H\) is the set of conjugates of \(H\); the stabilizer group of \(H\)
is the normalizer \(N_G(H):=\{g\in G|\hspace{1mm} gHg^{-1}=H\}\) of \(H\); and so the number of conjugates of
\(H\) is \([G:N_G(H)]\); we defined the kenrel of a group action; considered the group action
\(G\curvearrowright G/H\) to prove \({\rm cor}(H):=\cap_{g\in G} gHg^{-1}\) is the largest normal subgroup
of \(G\) which is a subset of \(H\) (this is called the normal core of \(H\)); we proved that
\(G/{\rm cor}(H)\) can be embedded into \(S_{G/H}\); used this result to prove: suppose \(H\) is a
subgroup of a finite group \(G\), and \([G:H]=p\) is the smallest prime factor of the order \(|G|\) of
\(G\). Then \(H\) is a normal subgroup of \(G\).
- Lecture 6: Here is my note for the sixth lecture.
In this lecture, first we defined a finite group \(G\) to be a \(p\)-group if its order is a power of
\(p\); assuming that \(G\) is a \(p\)-group, \(X\) is a finite set, and \(G\curvearrowright X\), we proved
that \(|X|\equiv |X^G| \pmod{p}\); using this result, we proved that, if \(G\) is a non-trivial \(p\)-group,
then its center \(Z(G)\) is non-trivial; assuming \(H\) is a \(p\)-subgroup
of \(G\) and \(p||G/H|\), we proved \(p||N_G(H)/H|\); we deduced that if \(H\) is a proper subgroup of a
\(p\)-group \(P\), then \(N_P(H)\neq H\); next we proved the Cauchy's theorem: if \(p\) is a prime factor of
\(|G|\), then there is \(g\in G\) such that \(o(g)=p\); finally we proved the first Sylow theorem:
if \(p^m| |G|\), then there are subgroups
\[P_1\unlhd P_2 \unlhd \cdots \unlhd P_m \leq G\]
such that \(|P_i|=p^i\) for \(1\leq i\leq m\).
- Lecture 7: Here is my note for the seventh lecture.
In this lecture, we defined Sylow \(p\)-subgroup of a finite group \(G\), and denoted the
set of all the Sylow \(p\)-subgroups of \(G\) by \({\rm Syl}_p(G)\); proved the second Sylow theorem:
if \(P_0\in {\rm Syl}_p(G)\) and \(Q\) is a \(p\)-subgroup of \(G\), then there is \(g\in G\) such that
\(g^{-1}Qg\subseteq P_0\); in particular the group action \(G\curvearrowright {\rm Syl}_p(G)\) (by conjugations)
is transitive; so \(|{\rm Syl}_p(G)|=[G:N_G(P_0)]\) if \(P_0\) is a Sylow \(p\)-subgroup of
\(G\); we deduced that \({\rm Syl}_p(N_G(P))=\{P\}\) if \(P\in {\rm Syl}_p(G)\); we used this result to prove
\(N_G(N_G(P))=N_G(P)\) if \(P\) is a Sylow \(p\)-subgroup; then we proved the third Sylow theorem:
\(|{\rm Syl}_p(G)|\equiv 1 \pmod{p}\); finally we used Sylow theorems to show a group
of order \(pq\) is not simple if \(p\) and \(q\) are distinct primes.
- Lecture 8: Here is my note for the eighth lecture.
In this lecture, we understood the possible group structures of a group \(G\) of order \(pq\);
in particular, we pointed out that, if \(p< q\) and \(p\nmid q-1\), then a group of order \(pq\) is cyclic. Next
we discussed the following problem: if \(G\) is a group of order \(p(p+1)\), then \(G\) has a normal
subgroup of order either \(p\) or \(p+1\). In the lecture note a much easier example is also mentioned:
a group of order \(p(p-1)\) has a normal subgroup of order \(p\). (During the lecture we discussed that,
if \(H\) and \(K\) are subgroups of \(G\), then the following is a bijection
\[H/(H\cap K) \rightarrow HK/K, \hspace{1mm} h(H\cap K) \mapsto h K\]
without a proof; and then deduced that \(|HK|=|H||K|/|H\cap K|\) if \(H,K\) are finite subgroups.)
- Lecture 9: Here is my note for the ninth lecture.
In this lecture, we defined an exact sequence, a short exact sequence, and a split short exact sequence;
we came up with the definition of a semi-direct product of two groups; and showed that if
\[1\rightarrow N\rightarrow G\rightarrow H\rightarrow 1\] is a split short exact sequence, then
\(G\simeq H\ltimes_c N\) for some group homomorphism \(c:H\rightarrow {\rm Aut}(N)\); using
this we gave an alternative way of describing group structures of groups of order \(pq\)
where \(p< q\) are primes: we recalled that any such group \(G\) admites a split short exact sequence
\[
1\rightarrow \mathbb{Z}/q\mathbb{Z} \rightarrow G \rightarrow \mathbb{Z}/p\mathbb{Z} \rightarrow 1;
\]
and so \(G\simeq \mathbb{Z}/p\mathbb{Z} \ltimes_c \mathbb{Z}/q\mathbb{Z}\) for some
\(c\in {\rm Hom}(\mathbb{Z}/p\mathbb{Z}, {\rm Aut}(\mathbb{Z}/q\mathbb{Z})) \). For any positive integer \(n\)
we have \({\rm Aut}(\mathbb{Z}/n\mathbb{Z})\simeq (\mathbb{Z}/n\mathbb{Z})^{\times}\), and for a prime \(q\) we have
\((\mathbb{Z}/q\mathbb{Z})^{\times}\simeq \mathbb{Z}/(q-1)\mathbb{Z}\); therefore
\[
{\rm Hom}(\mathbb{Z}/p\mathbb{Z}, {\rm Aut}(\mathbb{Z}/q\mathbb{Z}))\simeq {\rm Hom}(\mathbb{Z}/p\mathbb{Z}, \mathbb{Z}/(q-1)\mathbb{Z})
\]
which is non-trivial if and only if \(p|q-1\). Hence if \(p\nmid q-1\), then a group \(G\) of order \(pq\) is
isomorphic to \(\mathbb{Z}/p\mathbb{Z}\times\mathbb{Z}/q\mathbb{Z}\); and so by the Chinese Remainder Theorem
we get that \(G\) is cyclic. And if \(p|q-1\), then there are non-abelian groups of order \(pq\). Finally we mentioned
the statement of the Schur-Zassenhaus Theorem, and we deferred its proof to the next lecture.
- Lecture 10: Here is my note for the tenth lecture.
In this lecture, we restated the Schur-Zassenhaus theorem:
suppose \(N\) and \(H\) are two groups and \({\rm gcd}(|N|,|H|)=1\); then a
short exact sequence of the form
\(1\rightarrow N \rightarrow G \rightarrow H \rightarrow 1\)
splits; proved that to get the Schur-Zassenhaus theorem it is enough to show: if \(N\) is a normal subgroup
of \(G\) and \({\rm gcd}(|N|,[G:N])=1\), then there is a subgroup \(H\) of
\(G\) such that \(|H|=[G:N]\); next we made a few reductions to reduce proof of the
Schur-Zassenhaus Theorem to the case where \(N\) is abelian. The abelian case can be proved
using cohomology theory as it is outlined in your homework assignment.
- Lecture 11: Here is my note for the eleventh lecture.
In this lecture, we reviewed some of the facts about the symmetric group \(S_n\); for \(\sigma\in S_n\)
we defined \({\rm Fix}(\sigma):=\{i|\hspace{1mm} \sigma(i)=i\}\) and \({\rm supp}(\sigma):=\{1,2,\ldots,n\}\setminus {\rm Fix}(\sigma)\);
we said \(\sigma_1,\sigma_2\) are disjoint if \({\rm supp}(\sigma_1)\cap{\rm supp}(\sigma_2)=\varnothing\);
we proved two disjoint permutations commute; if \(\sigma_i\)'s are pairwise disjoint and \(X_i:={\rm supp}(\sigma_i)\), then
\((\sigma_1\cdots \sigma_m)|_{X_i}=\sigma_i|_{X_i}\); if \(X\) has at least 2 elements and \(\sigma_i\)'s are pairwise disjoint, then
\(X\) is an orbit of \(\langle \sigma_1\cdots\sigma_m\rangle\) exactly when \(X\) is an orbit of \(\sigma_i\) for some \(i\);
we used this to show the uniqueness of a cycle decomposition: suppose \(\sigma_i\)'s are disjoint cycles and \(\tau_j\)'s are disjoint cycles; then
\(\sigma_1\cdots \sigma_m=\tau_1\cdots \tau_k\) implies \(m=k\) and \(\sigma_1=\tau_{i_1},\ldots, \sigma_m=\tau_{i_m}\) where \(i_1,\ldots,i_m\) is a reordering of
\(1,\ldots, m\); finally the existence of a cycle decomposition is proved.
- Lecture 12: Here is my note for the twelfth lecture.
In this lecture, we further studied the symmetric group \(S_n\); how to find the order of an
element given its cycle type; conjugacy class of a permutation is determined by its cycle type;
cycle types and Young tableau; linking relation; transposition and the definition of the sign
function \(\epsilon:S_n\rightarrow \{\pm1\}\).
- Lecture 13: Here is my note for the thirteenth lecture.
In this lecture, we proved that the sign function \(\epsilon:S_n\rightarrow \{\pm1\}\) is a
non-trivial group homomorphism; defined odd and even permutations; mentioned the relation of
odd and even permutations with \(15\)-puzzle; defined the alternating group \(A_n\);
proved that \(3\)-cycles are conjugate in \(A_n\) if \(n\ge 5\); proved that a normal subgroup
of \(A_n\) which contains a \(3\)-cycle is the entire \(A_n\).
- Lecture 14: Here is my note for the fourteenth lecture.
In this lecture, we proved that the alternating group \(A_n\) is simple if \(n\ge 5\); along the way
we showed that, if \(N\) is a normal \(p\)-subgroup of a finite group \(G\), then
\(N\subseteq \bigcap_{P\in {\rm Syl}_p(G)} P\); we have also learned how to find the cycle type of
a permutation with a given order; for instance if \(o(\sigma)\) is a prime number \(p\), then
the cycle type of \(\sigma\) is of the form \(p\ge \cdots \ge p\ge 1\ge \cdots\ge 1\), and
the fact that it should be a partition of \(n\) can help us to get that the number of \(p\)'s is
at most \(\lfloor n/p\rfloor\).
In the note, I am including an example which can help you with Problem 5 of this week's homework assignment.
- Lecture 15: Here is my note for the fifteenth lecture.
In this lecture, we considered \(G\curvearrowright G\) by the left translations and let \(\phi:G\rightarrow S_G\)
be its associated group homomorphism; then discussed that for any \(g\in G\) the cycle type of \(\phi(g)\) is
\(o(g)\ge\dots\ge o(g)\) where the number of occurence of \(o(g)\) is \(|G|/o(g)\); used this fact to prove that a group of order \(2m\)
where \(m\) is odd has a normal subgroup of index \(2\). Next we proved the Jordan-Holder theorem: defined composition
series and composition factors of a finite group; proved the existence of a composition series and the uniqueness of
composition factors up to rearranging.
- Lecture 16: Here is my note for the sixteenth lecture.
In this lecture, we recalled what it means to say a subgroup \(H\) is generated by a non-empty subset \(X\)
of a group \(G\); We showed that, if \(\theta:G\rightarrow H\) is a group homomorphism and \(X\) is a non-empty
subset of \(G\), then \(\theta(\langle X\rangle)=\langle \theta(X)\rangle\); we defined the commutator subgroup \([H,K]\)
of two subgroups \(H,K\) of a group \(G\); proved that if \(\theta:G_1\rightarrow G_2\) is a group homomorphism and \(H,K\le G\), then
\(\theta([H,K])=[\theta(H),\theta(K)]\); proved that if \(H,K\) are normal subgroups of \(G\), then \([H,K]\) is a normal subgroup of
\(G\), and \([H,K]\subseteq H\cap K\); in particular, if \(\gcd(|H|,|K|)=1\) and \(H,K\unlhd G\), then \(H\) and \(K\) commute;
the derived series of a group was defined; we proved: suppose \(N\) is a normal subgroup of \(G\); then \(G/N\) is abelian
if and only if \([G,G]\subseteq N\); used this to show for a group \(G\) we have \(G^{(c)}=1\) if and only if there is a chain
\[
1=N_c\unlhd N_{c-1} \unlhd \cdots \unlhd N_0=G
\]
such that \(N_i/N_{i+1}\) is abelian; a group with these properties is defined to be solvable; proved that
if \(G\) is a solvable group, \(H\) is a subgroup, and \(N\) is a normal subgroup of \(G\), then \(H\) and \(G/N\) are solvable;
proved that a solvable simple group is a cyclic group of prime order.
- Lecture 17: Here is my note for the seventeenth lecture.
In this lecture, we proved that a finite group \(G\) is solvable if and only if all of its composition factors are
cyclic groups of prime order; we defined the lower and the upper central series: \(\gamma_1(G):=G\) and \(\gamma_{i+1}(G):=[G,\gamma_i(G)]\); and
\(Z_0(G):=\{1\}\) and \(Z_{i+1}(G)/Z_i(G):=Z(G/Z_i(G))\); we proved that, for any non-negative integer \(i\), \(\gamma_{i+1}(G)\) and \(Z_i(G)\) are
normal subgroups of \(G\); we have \(G=\gamma_1(G)\unrhd \gamma_2(G) \unrhd \cdots\) and \(1=Z_0(G)\unlhd Z_1(G) \unlhd \cdots \); We proved that
\(\gamma_{1+c}(G)=1\) if and only if \(Z_c(G)=G\); A group is called nilpotent if \(\gamma_{1+c}(G)=1\); we proved that if \(H\) is a proper subgroup
of a nilpotent group \(G\), then \(N_G(H)\neq H\).
- Lecture 18: Here is my note for the eighteenth lecture.
In this lecture, we proved that a finite group \(G\) is nilpotent if and only if it is a direct product of finite
\(p\)-groups; this is done in several steps: Step 1. In a finite nilpotent group any Sylow \(p\)-subgroup is normal;
Step 2. If \(N_1,\ldots,N_k\unlhd G\) and \(\gcd(|N_i|,|N_j|)=1\) for \(i\neq j\), then
\((x_1,\ldots,x_k)\mapsto x_1\dots x_k\) is an isomorphism from \(N_1\times \cdots \times N_k\) to \(N_1\cdots N_k\);
Step 3. A finite \(p\)-group is nilpotent; Step 4. If \(G_1,\ldots,G_k\) are nilpotent groups, then
\(G_1\times \cdots \times G_k\) is nilpotent. Next we proved that a finite group \(G\) is nilpotent if and only if
all of its maximal subgroups are normal.
- Lecture 19: Here is my note for the nineteenth lecture.
In this lecture, we proved the following properties of a nilpotent group \(G\):
(1) \(G\) is solvable; (2) If \(H\le G\) and \(N \unlhd G\), then \(H\) and \(G/N\) are nilpotent; (3)
If \(N\) is a non-trivial normal subgroup of \(G\), then \(Z(G)\cap N\neq 1\). Then we defined the Frattini subgroup
\(\Phi(G)\) of a group \(G\); that is the intersection of all the maximal subgroups of \(G\). We observed that the
Frattini subgroup \(\Phi(G)\) is a characteristic subgroup of \(G\). Then we proved that if \(G\) is a finite \(p\)-group,
then \(\Phi(G)=G^p[G,G]\).
- Lecture 20: Here is my note for the twentieth lecture.
In this lecture, first we defined the language \(\mathcal{L}(X)\) that is given by a non-empty set \(X\) (called the alphabet); we showed that
\(\mathcal{L}(X)\) is the free monoid generated by \(X\); that means it satisfies certain universal property. Then we defined the free product
\(\ast_{i\in I} G_i\) of a family of groups \(\{G_i\}_{i\in I}\), and proved that it satisfies certain universal property; at the end we defined
the free group generated by a non-empty set \(X\) to be \(\ast_{x\in X} \mathbb{Z}\).
(I am using Topics in Geometric Group Theory, by Pierre de la Harpe, for this part of the course.)
- Lecture 21: Here is my note for the twenty first lecture.
In this lecture, the universal property of free product can be found in the notes; we discussed reduced words in \(\mathcal{L}(X)\), where
\(X\) is the disjoint union of a family of groups, and reduced and cyclically reduced words in \(\mathcal{L}(X\cup X^{-1})\). We proved the following
version of the 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 \(\langle G_1\cup G_2\rangle\simeq G_1\ast G_2\).
; Then we mentioned the action of \({\rm SL}_2(\mathbb{R})\) on the projective space \(P(\mathbb{R}^2)\), and analyzed the set of fixed points of
\(\left[\begin{smallmatrix}1&2\\0&1\end{smallmatrix}\right]\) and \(\left[\begin{smallmatrix}1&0\\2&1\end{smallmatrix}\right]\); and we said how to use the mentioned action and the
Ping-Pong Lemma to show that \(\left[\begin{smallmatrix}1&2\\0&1\end{smallmatrix}\right]\) and \(\left[\begin{smallmatrix}1&0\\2&1\end{smallmatrix}\right]\) freely generate a subgroup of
\({\rm SL}_2(\mathbb{Z}) \).
- Lecture 22: Here is my note for the twenty second lecture.
In this lecture, we saw several applications of the ping-pong lemma: finished the proof of why
\(\left[\begin{smallmatrix}1&2\\0&1\end{smallmatrix}\right]\) 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.
Along the way we discussed a bit more general Schottky groups. At the end of the lecture, Tits alternative was stated.
- Lecture 23: Here is my note for the twenty third lecture.
In this lecture, we first proved that a virtually solvable group does not have a non-commutative free subgroup; presentation of group
was defined; proved \(\langle x|x^n \rangle\simeq \mathbb{Z}/n\mathbb{Z}\) and \(D_{2n}\simeq \langle x,y| x^2, y^n, xyx^{-1}=y^{-1}\rangle\); proved that
\(\langle\overline{\left[\begin{smallmatrix}1&2\\0&1\end{smallmatrix}\right]},\overline{\left[\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right]}\rangle\simeq \langle x,y|y^2\rangle\).
- Lecture 24: Here is my note for the twenty fourth lecture.
In this lecture, we defined matrix ring, group ring (monoid ring), polynomial ring, Banach algebra of integrabale functions, ring homomorphism,
the evaluation map of ring of polynomials; and we emphasized on the distinction between polynomials and the functions that are given by those.
- Lecture 25: Here is my note for the twenty fifth lecture.
In this lecture, we recalled the quotient ring, the first isomorphism theorem, zero-divisor, integral domain, field,
degree function on the ring of polynomials, and find the units of ring of polynomials with coefficients in an integral domain.
- Lecture 26: Here is my note for the twenty sixth lecture.
In this lecture, we defined the definitions of prime and maximal ideals; proved \(\mathfrak{p}\unlhd A\) is a prime ideal if and only if
\(A/\mathfrak{p}\) is an integral domain; \(\mathfrak{m}\unlhd A\) is a maximal ideal if and only if \(A/\mathfrak{m}\) is a field; deduced that
\({\rm Max}(A)\subseteq {\rm Spec}(A)\); defined Partially Ordered Set (poset), chain, upper-bound, maximal element; stated Zorn's lemma and the Axiom
of Choice; Stated the following theorem: Suppose \(\mathfrak{a} \unlhd A\), \(S\subseteq A\) is a multiplicatively closed subset, and \(\mathfrak{a}\cap S=\varnothing\).
Let \(\Sigma_{\mathfrak{a},S}:=\{\mathfrak{b}\unlhd A|\hspace{1mm} \mathfrak{a}\subseteq \mathfrak{b}, \mathfrak{b}\cap S=\varnothing\}\).
Then (1) \(\Sigma_{\mathfrak{a},S}\) has a maximal element, (2) A maximal element of \(\Sigma_{\mathfrak{a},S}\) is a prime ideal. Then we deduced two corollaries:
If \(\mathfrak{a} \unlhd A\), \(S\subseteq A\) is a multiplicatively closed subset, and \(\mathfrak{a}\cap S=\varnothing\), then there is \(\mathfrak{p}\in {\rm Spec}(A)\)
such that \(\mathfrak{a}\subseteq \mathfrak{p}\) and \(S\cap \mathfrak{p}=\varnothing\). And If \(\mathfrak{a}\) is a proper ideal of \(A\),
then there is \(\mathfrak{m}\in {\rm Max}(A)\) such that \(\mathfrak{a}\subseteq \mathfrak{m}\).
- Lecture 27: Here is my note for the twenty seventh lecture.
In this lecture, we proved that Suppose \(\mathfrak{a} \unlhd A\), \(S\subseteq A\) is a multiplicatively closed subset, and \(\mathfrak{a}\cap S=\varnothing\).
Let \(\Sigma_{\mathfrak{a},S}:=\{\mathfrak{b}\unlhd A|\hspace{1mm} \mathfrak{a}\subseteq \mathfrak{b}, \mathfrak{b}\cap S=\varnothing\}\).
Then (1) \(\Sigma_{\mathfrak{a},S}\) has a maximal element, (2) A maximal element of \(\Sigma_{\mathfrak{a},S}\) is a prime ideal, which was stated in the previous lecture;
defined the nilradical \({\rm Nil}(A)\) of a ring \(A\); proved that \({\rm Nil}(A)=\bigcap_{\mathfrak{p}\in {\rm Spec}(A)}\mathfrak{p}\); defined Euclidean Domain and
Principal Ideal Domain (PID); proved that a Euclidean Domain is a PID; mentioned that the ring of integers \(\mathbb{Z}\) and the ring \(F[x]\) of polynomials over a field \(F\)
are Euclidean domains; we will prove in the next lecture that the ring of Gaussian integers \(\mathbb{Z}[i]:=\{a+bi|\hspace{1mm} a,b\in \mathbb{Z}\}\) is a Euclidean Domain.
- Lecture 28: Here is my note for the twenty eighth lecture.
In this lecture, we defined irreducible and prime elements of an integral domain; proved suppose \(D\) is an integral domain and \(a\) is a
non-zero element of \(D\), then \(a\) is prime if and only if \(\langle a\rangle\) is prime; \(a\) is irreducible if and only if \(\langle a\rangle\) is maximal among all the
proper principal ideals of \(D\); \(a\) is prime implies that \(a\) is irreducible; \(b,c\in D\) are associates, i.e. \(b\in cD^{\times}\) if and only
if \(\langle b\rangle=\langle c\rangle\). Proved that in a PID, a non-zero element is prime if and only if it is irreducible.
Defined a Noetherian ring; mentioned that \(A\) is Noetherian if and only if \(A\) satisfies the ascending chain condition (acc);
proved that \(A\) is Noetherian if and only if any ideal of \(A\) is finitely generated; deduced that a PID is Noetherian; said what it means to say an
integral domain is a Unique Factorization Domain (UFD); mentioned how to use the Noetherian property of a PID to show why any non-zero non-unit element of a PID can be written
as a product of irreducibles (look at the lecture note for the details of the argument); mentioned how to use the fact that in a PID an irreducible element in is a prime to
deduce the uniqueness of irreducible decomposition of non-zero non-unit elements of a PID. And concluded that a PID is a UFD . In the lecture note I have included a proof
of why \(\mathbb{Z}[i]\) is a Euclidean domain. So overall we have:
\[
\text{ Euclidean Domain } \Rightarrow \text{ PID } \Rightarrow \text{ UFD }.
\]
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