| My notes.
You can use my lecture notes from 2018.
I will summarize what gets covered during the lectures here. You should go over the notes and your book on the relevant topics.
- 9/26.
Group of symmetries of an object. Symmetric group and it dependence on the cardinality of the set. Group of automorphisms of a graph. Cayley graph.
Group of automorphisms of an \(n\)-cycle, and dihedral groups. Group generated by a subset. Order of an element. Conjugation and inner automorphisms.
Normal subgroup. Important relations of the standard generators of a dihedral group. Group of isometries of the Euclidean plane. Group action. The
meta example of the group of symmetries of an object \(X\) acting on \(X\).
- 10/1.
Various examples of group actions: \(S_n\curvearrowright \{1,\ldots,n\}\),
\({\rm Aut}(G)\curvearrowright G\), \({\rm GL}_n(\mathbb{R})\curvearrowright \mathbb{R}^n\).
More abstract examples: \(G\curvearrowright G/H\) by left translations, \(G\) acts on the set of all subgroups
of \(G\) via conjugation, \(G\) acts on the set of all subgroups of finite index of \(G\) via conjugation.
Induced group action:
if \(\theta:G\to H\) is a group homomorphism and \(H\curvearrowright X\), we get an action of \(G\) on \(X\).
Recalled that \(c:G\to {\rm Aut}(G), c(g):=c_g\) where \(c_g(x):=gxg^{-1}\) is a group homomorphism. The kernel of this map
is the center of \(G\) and the image of \(c\) is called the group of inner automorphisms of \(G\). Hence, \({\rm Inn}(G)\simeq G/Z(G)\).
Used the induced action to obtain an action \(G\curvearrowright G\) via conjugation.
We gave bijections between the set of actions of \(G\) on \(X\) and the set of group homomorphisms from
\(G\) to \(S_X\). We showed how a special case of this result implies the Cayley theorem: every group \(G\) can be
embedded in the symmetric group \(S_G\). Defined the stabilizer subgroup of \(G\) associated with \(x\). Proved the orbit-stabilizer theorem:
\(G_x\) is a subgroup and
\(G/G_x\to G\cdot x, gG_x\mapsto g\cdot x\) is a bijection.
- 10/3.
Suppose \(G\curvearrowright X\). Then \(G\cdot x= G\cdot y\) if and only if \(y\in G\cdot x\) if and only if
\(G\cdot x\cap G\cdot y \neq \varnothing\). Let \(_G\setminus^X:=\{G\cdot x\mid x\in X\}.\)
Then \(_G\setminus^X\)
is a partition of \(X\) and it is called the quotient of \(X\) by \(G\). Let \(X^G\) be the set of \(G\)-fixed points. Then
\(|X|=|X^G|+\sum_{G\cdot x\in _G\setminus^X, |G\cdot x|>1} [G:G_x]\). Considered the action \(G\curvearrowright G\)
via conjugation. Observed that the \(G\)-orbit of \(x\) is the conjugacy class of \(x\), and argued why the staibizer subgroup
of \(G\) associated with \(x\) is the centralizer \(C_{G}(x)\) of \(x\) in \(G\). Hence,
\(|{\rm Cl}(x)|=[G:C_G(x)]\). Deduced the class equation. Defined the kernel of a group action. Proved that if
\(N\) is the kernel of a group action \(G\curvearrowright X\), then \(G/N\hookrightarrow S_X\). Considered the action
\(G\curvearrowright G/H\) via left translations, and proved that this has only one orbit (we say such an action is transitive).
Noticed that the kernel of this action is \({\rm cor}(H):=\cap_{x\in G} xHx^{-1}\). This is called the normal core of \(H\) in \(G\).
We proved that \({\rm cor}(H)\unlhd G\), \({\rm cor}(H)\leq H\), \([G:{\rm cor}(H)]|[G:H]!\), and if \(N\) is a normal subgroup
of \(G\) and \(N\leq H\), then \(N\leq {\rm cor}(H)\). Used this to prove that if \(H\leq G\) and \([G:H]\) is a smallest
prime factor of \(|G|\), then \(H\) is a normal subgroup of \(G\). Next, we showed that \(G_{g\cdot x}=g G_x g^{-1}\), in particular
\(|G_{x}|=|G_{y}|\) if \(G\cdot x=G\cdot y\); this means \(|G_x|\) only depends on the orbit \(G\cdot x\). We also proved that
\(h\cdot X^g=X^{hgh^{-1}}\) where \(X^g:=\{x\in X\mid g\cdot x=x\}\). This was used to prove
\(
|_G\setminus^X|=\frac{1}{|G|}\sum_{g\in G} |X^g|.
\)
- 10/8.
Pointed out that \(G\) acts on the set \({\rm Sub}(G)\) of all subgroups of \(G\) via conjugation. The stabilizer subgroup of \(G\) associated
with \(H\in {\rm Sub}(G)\) is the normalizer of \(H\) in \(G\), it is denoted by \(N_G(H)\) and we have \(N_G(H)=\{g\in G\mid gHg^{-1}=H\}\). By the
orbit-stabilizer theorem, we deduce that the number of conjugates of \(H\) is equal to \([G:N_G(H)]\).
Next, we showed the following fundamental result: suppose \(P\) is a group of order \(p^n\) where \(p\) is prime and \(n\) is a positive integer.
Suppose \(P\) acts on a finite set \(X\). Then \(|X|\equiv |X^G|\pmod{p}\) where \(X^G:=\{x\in X\mid \forall g\in G, g\cdot x=x\}\).
We used this result to prove that if \(|P|=p^n\) and \(N\) is a non-trivial normal subgroup of \(P\), then \(Z(P)\cap N\neq 1\).
Next, we proved that if
\(p\) is a prime factor of the order of a finite group \(G\), then the number of elements of \(G\) that are of order \(p\) is congruent to \(-1\) modulo \(p\); in particular
there is an element of order \(p\) (known as Cauchy's theorem).
Using Cauchy's theorem, we proved that if \(G\) is a finite \(p\)-group, then the order \(|G|\) of \(G\) is a power of \(p\).
Next, we proved that if \(P\) is a subgroup of a finite group \(G\) and \(|P|=p^n\), then \(|G/P|\equiv |N_G(P)/P| \pmod{p}\); in particular, if \(p| |G/P|\), then
\(p| |N_G(P)/P|\).
Using the previous result and Cauchy's theorem, by induction on \(k\), we proved Sylow's first theorem: suppose \(p^k\) divides the order of a finite group \(G\). Then there is a chain
of subgroups \(P_1\leq P_2\leq \cdots \leq P_k\leq G\) such that \(|P_i|=p^i\) for all \(1\leq i\leq k\).
We say a subgroup \(P\) of \(G\) is a Sylow \(p\)-subgroup if \(|P|\) is a power of \(p\) and \(p\nmid |G/P|\). The set of all Sylow \(p\)-subgroups of \(G\) is denoted by
\({\rm Syl}_p(G)\). By Sylow's first theorem, we have that \({\rm Syl}_p(G)\) is not empty.
Next, we proved Sylow's second theorem: suppose \(Q\) is a \(p\)-subgroup of a finite group \(G\) and \(P_0\in {\rm Syl}_p(G)\), then there exists \(x\in G\) such that \(Q\subseteq xP_0x^{-1}\);
in particular, every two Sylow \(p\)-subgroups are conjugate of each other. Alternatively, we can say that \(G\) acts transitively on \({\rm Syl}_p(G)\) via conjugation.
We also obtained that \(|{\rm Syl}_p(G)|=[G:N_G(P_0)]\).
Using Sylow's second theorem, we deduced that \({\rm Syl}_p(N_G(P))=\{P\}\) if \(P\in {\rm Syl}_p(G)\). Used this to show that for every automorphism \(\theta\) of \(G\),
\({\rm Syl}_p(\theta(N_G(P)))=\{\theta(P)\}\). Hence, \(\theta(N_G(P))=N_G(P)\) implies that \(\theta(P)=P\).
- 10/10.
Proved that \(N_G(N_G(P))=N_G(P)\) if \(P\) is a Sylow \(p\)-subgroup of \(G\). Proved Sylow's 3rd theorem. Showed that a group of order \(pq\), where \(p,q\) are two primes and \(p\) is less than \(q\), has a normal Sylow \(q\)-subgroup.
We showed that if \(p\nmid q-1\), \(p,q\) are two primes and \(p\) is less than \(q\), then a group of order \(pq\) is cyclic. We showed that every group \(G\) of order \(pq\) is of them form \(PQ\) where \(P\) is a Sylow \(p\)-subgroup and
\(Q\) is the unique Sylow \(q\)-subgroup. To show this, we proved that if \(H\) and \(K\) are two subgroups, then the following is a well-defined bijection \(H/(H\cap K)\to HK/K,\quad h(H\cap K)\mapsto hK\); deduced that
\(|HK|=\frac{|H||K|}{|H\cap K|}\). To understand the group structure of a group of order \(pq\) further, we noticed that \(P\) acts on \(`Q\) via conjugation, and so we get a group homomorphism from \(P\) to \({\rm Aut}(Q)\).
Then we proved that \({\rm Aut}(C_n)\simeq (\mathbb{Z}/n\mathbb{Z})^\times\) where \(C_n\) is a cyclic group of order \(n\).
-
No class on 10/15 there will be a make-up lecture on Wednesday 10/23 from 9:30am-10:50am in APM 6402.
- 10/17.
Showed that every group of order \(pq\) where \(p\) and \(q\) are distinct primes is of the form \(\{g^i h^j\mid 0\leq i\leq p-1, 0\leq j\leq q\}\) such that the conjugation of \(Q=\langle h\rangle\) by \(P=\langle g\rangle\)
is given via a group homomorphism \(f: P\to {\rm Aut}(Q)\). Next, we showed every group of order \(p(p-1)\) has a normal subgroup of order \(p\). Then we proved that every group of order \(p(p+1)\) has a normal subgroup of
order either \(p\) or \(p+1\). After that, we defined exact sequences, short exact sequences (SES), standard SES, homomorphisms of SES, splitting SES. Proved that every SES is isomorphic to a standard SES.
- 10/22.
Showed that a standard SES \(1\to N\to G\to G/N\to 1\) splits if and only if there exists a subgroup \(H\) of \(G\) such that \(NH=G\) and \(N\cap H =\{1\}\) (we say \(H\) is transverse to \(N\)).
Proved that if \(G\) is a finite group, then a standard SES splits if and only if there exists a subgroup \(H\) of \(G\) such that \(N\cap H=\{1\}\) and \(|H|=|G/N|\). Defined the semidirect product
of two groups \(N\) and \(H\) with respect to a group homomorphism \(f:H\to {\rm Aut}(N)\). This is denoted by \(N\rtimes_f H\). We showed that
\(1\to N\xrightarrow{i} N\rtimes_f H\xrightarrow{p} H\to \) where \(i(n):=(n,1)\) and \(p(n,h):=h\) is a splitting SES (we refer to this as the standard SES of a semidirect product). Proved that
Proved that if \(1\to N\to G\to H\to 1\) is a splitting SES, then there are a group homomorphism \(f:H\to {\rm Aut}(N)\) and a group isomorphism \(\theta:G\to N\rtimes_f H\) such that
\(({\rm id}_N, \theta, {\rm id}_H)\) is an isomorphism of SES between \(1\to N\to G\to H\to 1\) and the standard SES of \(N\rtimes_f H\).
- 10/23.
Completed the proof of the characterization of splitting SES in terms of standard SES of a semidirect product. Showed how this can help us get a better understanding of groups of order
\(pq\). Mentioned the Schur-Zassenhaus theorem, and proved the needed reductions. The completion is done in your HW assignment.
-
10/24.
Discussed symmetric groups. Cycle decomposition of permutation. The cycle type of a permutation. Conjugacy classes: two permutations are conjugate if and only if they have the same
cycle types. The number of conjugacy classes of \(S_n\) is the same as the number of partitions of \(n\). Linking relations. Transposition, and why every permutation can be written as a product
of transpositions.
-
10/29.
\(S_n\) acts on the ring of polynomials \(\mathbb{Z}[x_1,\ldots,x_n]\) via \((\sigma\cdot f)(x_1,\ldots,x_n)=f(x_{\sigma(1)},\ldots,x_{\sigma(n)})\). Noticed that \(f\mapsto \sigma\cdot f\) is an automorphism of
\(\mathbb{Z}[x_1,\ldots,x_n]\). Defined \(\Delta(x_1,\ldots,x_n):=\prod_{i\lt j}(x_i-x_j)\). Then \((\sigma\cdot \Delta)^2=\Delta^2\).
(We had our midterm).
-
10/31.
Showed that \(\Delta^2=(-1)^{n(n-1)/2} \prod_{i \neq j} (x_i-x_j)\). Deduce that there is a function \(\epsilon:S_n\to \{\pm 1\}\) such that \(\sigma\cdot \Delta=\epsilon(\sigma) \Delta\).
Use the properties of action,
to show that \(\epsilon:S_n\to \{\pm 1\}\) is a group homomorphism. Showed that \(\epsilon(\sigma)=(-1)^{n_{\sigma}}\) where \(n_{\sigma}\) is the crossing number of \(\sigma\). Showed that the crossing number of
a transposition is odd, and so \(\epsilon((i\hspace{1mm} j))=-1\). Defined odd and even perutations. Proved that \(A_n\) is simple if \(n\geq 5\). Defined the composition series of a finite group, and proved its existence. Stated the
Jordan-Holder theorem.
-
11/5.
Proved the uniqueness of composition factors. Metioned that the composition factors of
a finite abelian group are cyclic groups of prime order. Next, we provided a description of finite groups
whose composition factors are all cyclic groups of prime order. To do so, we defined the derived series subgroups, and
solvable groups. Proved that \(G^{(k)}=\{1\}\) if and only if there exists a sequence of
subgroups \(\{1\}=N_k\trianglelefteq N_{k-1} \trianglelefteq \cdots \trianglelefteq N_0=G\) such that
\(N_i/N_{i+1}\) is abelian for all \(i\). Proved that a subgroup and a quotient of a
solvable group is solvable. Showed that if \(\phi:G_1\to G_2\) is a group homomorphism and \(H,K\) are
two subgroups of \(G_1\), then \(\phi([H,K])=[\phi(H),\phi(K)]\); in particular,
\(\phi(G^{(k)})=\phi(G)^{(k)}\) for all \(k\).
-
11/7.
Showed that a solvable simple group is a cyclic group of prime order.
Showed that a finite group is solvable if and only all of its composition factors are cyclic groups
of prime order. Defined central series: \(N_1 \leq \cdots \leq N_k\) is called a central series of \(G\) if
\(N_i\trianglelefteq G\) for all \(i\) and \(N_{i+1}/N_i\subseteq Z(G/N_i)\) for all \(i\). Proved that
\(N_{i+1}/N_i\subseteq Z(G/N_i)\) if and only if \([G,N_{i+1}]\subseteq N_i\). Defined upper and lower central
series. Proved that they are characteristic subgroups and form central series. Proved for some non-negative
integer \(c\), that following are equivalent:
(a) \(\gamma_{c+1}(G)=\{1\}\), (b) \(Z_c(G)=G\), (c) there exists a central series
\(
\{1\}=:N_0\trianglelefteq \cdots \trianglelefteq N_c:=G.
\)
A group \(G\) is called nilpotent with nilpotency \(c\) if the above above properties hold.
Proved that a subgroup and a quotient of a nilpotent group is nilpotent. Pointed out every abelian Group
is nilpotent, \(D_{\infty}\) has an abelian normal subgroup \(A\) such that \(D_{\infty}/A\) is abelian, but
\(Z(D_{\infty})=\{1\}\) and so \(D_{\infty}\) is not nilpotent.
-
11/12.
Proved that if \(G/Z(G)\) is nilpotent, then \(G\) is nilpotent. Showed that a finite \(p\)-group is nilpotent.
Proved that if \(N_1\leq \cdots \leq N_k\) is a central series of \(G\), then the following holds:
- For every subgroup \(H\) of \(G\), \(N_i\subseteq H\) implies that \(N_{i+1}\subseteq N_G(H)\).
- For every normal subgroup \(N\) of \(G\), \( [N,(N\cap N_i)] \subseteq (N\cap N_{i+1})\).
Used this to deduce that if \(G\) is nilpotent and \(H\) is a proper subgroup of \(G\), then \(H\) is a proper subgroup of \(N_G(H)\). And if \(G\) is a nilpotent and \(N\) is a
non-trivial normal subgroup, then \(N\cap Z(G)\) is non-trivial.
Next, we showed that for a finite group the following statements are equivalent:
- \(G\) is nilpotent.
- For every prime \(p\), \(G\) has a unique Sylow \(p\)-subngroup.
- \(G\simeq \prod_{i=1}^k P_i\) where \(P_i\) is a finite \(p_i\)-group and \(p_i\)'s are distinct primes.
-
11/14.
Showed that a finite group \(G\) is nilpotent if and only if every maximal subgroup of \(G\) is normal. Defined a monoid. Defined the free object in a category.
Proved the language \(\mathcal{L}(X)\) in the alphabet \(X\) is the free monoid generted by \(X\). Defined the free product of a family of groups and proved its universal property:
there is a natural bijection between the set \(\prod_{i\in I} {\rm Hom}(G_i,G)\) and \({\rm Hom}(\ast_{i\in I} G_i,G)\) (in category theory, \(\ast_{i\in I} G_i\) is called the co-product of \(G_i\)'s).
-
11/19.
Showed that a group generated by two elements of order 2 is a quotient of \(\mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}\). Proved a Ping-Pong Lemma, and used it to show that
the group generted by \(\begin{pmatrix}1& 2\\0& 1 \end{pmatrix}\) and \(\begin{pmatrix}1& 0\\2& 1 \end{pmatrix}\) is isomorphic to \(\mathbb{Z}\ast \mathbb{Z}\). Pointed out that the same argument shows that
the group generated by \(\begin{pmatrix}1& z\\0& 1 \end{pmatrix}\) and \(\begin{pmatrix}1& 0\\z& 1 \end{pmatrix}\) for every \(z\in \mathbb{C}\) with \(|z|\geq 2\) is isomorphic to \(\mathbb{Z}\ast \mathbb{Z}\).
Defined a residually finite group and proved that \(\mathbb{Z}\ast \mathbb{Z}\) is residually finite.
Briefly pointed out that for every transcendental \(z\) the same statement holds.
-
11/21.
Proved that the group generated by the following elements of \({\rm PSL}_2(\mathbb{Z})\), \(\pm \begin{pmatrix}1& 2\\0& 1 \end{pmatrix}\) and \(\pm \begin{pmatrix}0& 1\\-1& 0 \end{pmatrix}\) is isomorphic to
\(\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}\). Defined the free group \(F(X)\) generated by a set \(X\) and proved its universal property. Proved the uniquness of a free object if it exists.
Proved that a non-cyclic free group is not solvable. Mentioned the Tits alternative theorem.
Defined the group given by a presentation \(\langle X \mid R\rangle\). Proved that \(\langle x\mid x^n\rangle\simeq \mathbb{Z}/n\mathbb{Z}\). Presented a general method of approaching this type of question.
Used this method to show that
\(\langle a,b \mid aba^{-1}=b^{-1}, a^2, b^n\rangle\simeq D_{2n}\).
-
11/26.
Monoid ring \(RM\) where \(R\) is a ring and \(M\) is a monoid. Viewing this ring as functions \(f:M\to R\) with finite support and defining the covolution of two such functions. Pointing out the
Banach algebra \((\ell^1(M),+,\ast)\). Remarking that all the functions \(f:\mathbb{N}\to \mathbb{C}\) under pointwise addition and convolution form a ring. Remarking that the monoid ring of the monoid
\((\mathbb{Z}^{\geq 0})^n\) over the ring \(R\) is isomorphic to the ring of polynomials \(R[x_1,\ldots,x_n]\). Discussed that if \(R\) is a subring of \(S\), \(\mathbf{s}:=(s_1,\ldots,s_n)\), and
\(s_1,\ldots,s_n\in Z(S)\), then the evalutaion map \(e_{\mathbf{s}}:R[x_1,\ldots, x_n]\to S,\quad e_{\mathbf{s}}(f):=f(\mathbf{s})\) is a ring homomorphism. Defined the kernel of a ring homomorphism. Defined an ideal of
a ring. Defined the quotient ring.
-
12/03.
Proved the first isomorphism theorem for rings. Defined the divisibility of elements and ideals. Defined zero-divisors, units, and integral domains. Defined fields. Pointed out that a unit is never a zero-divisor, and
deduced that a field is an integral domain. Proved that a finite integral domain is a field. Defined the leading term, the leading coefficient, and the degree of a polynomial. Proved that if \(D\) is an integral domain, then
\({\rm Ld}(fg)={\rm Ld}(f){\rm Ld}(g)\) and \(\deg fg=\deg f+\deg g\). Deduced that if \(D\) is an integral domain, then \(D[x]^\times=D^\times\) and \(D[x]\) is an integral domain. Defined a nilpotent element.
Proved that if \(a\in {\rm Nil}(A)\), then \(1-a\in A^\times\). Defined prime and maximal ideals and proved that
\(\mathfrak{p}\) is a prime ideal of \(A\) if and only if \(A/\mathfrak{p}\) is an integral domain. And \(\mathfrak{m}\) is a maximal ideal of \(A\) if and only if \(A/\mathfrak{m}\) is a field.
-
12/05.
Defined the set of prime divisors of an ideal \(V(\mathfrak{a})\). Mentioned that these sets can be considered as the closed subsets of \({\rm Spec}(A)\), and it defines a topology which is called the Zariski-topology.
Defined a POSet, a chain, a maximal element, an upper-bound. Stated Zorn's lemma. Proved the union of a chian of ideals is always an ideal. Pointed out the same argument works
for almost all algebraic structures: vector spaces, groups, etc. Proved that the following statement holds: suppose \(\mathfrak{a}\) is an ideal \(A\) and \(S\) is multiplicatively closed subset of \(A\). Suppose
\(\mathfrak{a}\cap S=\varnothing\). Let \(\Sigma_{\mathfrak{a},S}:=\{\mathfrak{b}\mid \mathfrak{b}|\mathfrak{a}, \mathfrak{b}\cap S=\varnothing\}\). Then (1) \(\Sigma_{\mathfrak{a},S}\) has a maximal element, (2)
every maximal element of \(\Sigma_{\mathfrak{a},S}\) is a prime ideal, and (3) every maximal element of \(\Sigma_{\mathfrak{a},\{1\}}\) is a maximal ideal of \(A\). We deduced the following statements:
in the above setting, if \(\mathfrak{a}\cap S=\varnothing\), then there is \(\mathfrak{p}\in V(\mathfrak{a})\) such that \(\mathfrak{p}\cap S=\varnothing\). We also showed that every proper ideal has a maximal divisor.
Next, we showed that \({\rm Nil}(A)=\bigcap_{\mathfrak{p}\in {\rm Spec}(A)} \mathfrak{p}\). Last, we defined irreducible and prime elements, and what it means for two elements to be associates. Proved that \(a\) is irreducible
if and only if \(a\neq 0\) and \(\langle a \rangle\) is maximal among proper principal ideals. Proved that \(a\) and \(b\) are associates if and only if \(\langle a\rangle=\langle b\rangle\). Menioned that \(a\) is
prime if and only if \(a\neq 0\) and \(\langle a\rangle\) is a prime ideal.
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