Date
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What we discussed/How we spent our time
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Aug 28
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Prerequisites.
Syllabus. Text.
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Aug 30
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We discussed Lagrange's Theorem, Cauchy's Theorem (2 proofs),
and Sylow's First Theorem (2 proofs).
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Sep 1
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We defined the commutator, and discussed abelianness and solvability.
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Sep 6
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We discussed this handout.
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Sep 8
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We discussed verbal and marginal subgroups,
and solvable groups.
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Sep 11
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We discussed H, S, P properties of the
commutator and started discussing
nilpotent groups.
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Sep 13
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We showed that the descending central series and the ascending
central series of a nilpotent group
have the same length, and that this length
is a lower bound on the length of any central series.
We discussed $T(n,\mathbb F)$ and
$U(n,\mathbb F)$, and explained why
$U(n,\mathbb F)\lhd T(n,\mathbb F)$,
$T(n,\mathbb F)/U(n,\mathbb F)$ $\cong$ $(\mathbb F^{\times})^n$,
$U(n,\mathbb F)$ is nilpotent and $T(n,\mathbb F)$ is not,
and, if $|\mathbb F| = p^k$, then
$U(n,\mathbb F)$ is a Sylow $p$-subgroup of $T(n,\mathbb F)$.
We discussed commutator collection for nilpotent groups.
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Sep 15
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We discussed periodicity, local finiteness,
varieties of groups, Burnside's problem,
and consequences of a commutator collection.
We began a proof of a theorem characterizing
finite nilpotent groups in several ways.
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Sep 18
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We finished the proof of a theorem characterizing
finite nilpotent groups. We defined
radicals and residues, the Fitting subgroup
and the Frattini subgroup. We proved
Fitting's Theorem. We showed that
if $G$ is a finite solvable group,
then $C_G(\textrm{Fit}(G))\leq \textrm{Fit}(G)$,
hence there is a short exact sequence $1\to Z(F)\to G\to G/Z(F)\to 1$
with $G/Z(F)$ embeddable in $\textrm{Aut}(F)$. This yields
a finite-to-one correspondence between finite solvable groups and
finite nilpotent groups.
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Sep 20
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$\Phi(G)$ is the set of nongenerators,
and also the intersection of maximal subgroups of $G$.
We computed $\Phi(G)$ for small groups.
We explained why $\Phi(G)$ is independent
of $\textrm{Fit}(G)$ in general, but
$\Phi(G)\leq \textrm{Fit}(G)$ for finite groups.
We gave most of the proof that
$\textrm{Fit}(G)/\Phi(G)$ is a semisimple
abelian group when $G$ is finite.
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Sep 22
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We discussed modular lattices, and proved that
the modular law holds in subgroup
lattices in the following form:
If $A, B, C\leq G$, $A\leq B$, and $AC=CA$, then
$B\cap (AC) = A(B\cap C)$. We used this to prove
that if $N\lhd G$, then $\Phi(N)\lhd \Phi(G)$.
We proved the Burnside Basis Theorem. We stated the Dedekind-Baer
Theorem on groups whose subgroups are all normal.
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Sep 25
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We discussed some elements of the proof of
the Dedekind-Baer
Theorem on groups whose subgroups are all normal.
We defined regular $p$-group, and proved
that a 2-step nilpotent group satisfies
$(xy)^n = x^ny^n[x,y]^{-\binom{n}{2}}$.
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Sep 27
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We discussed special and extraspecial $p$-groups.
We explained that if $G$ is extraspecial, then
the commutator induces a symplectic form
$[,]:G/G'\times G/G'\to Z(G)$. We stated
the classification of finite dimensional
symplectic forms, and used this to deduce
that $G$ has subgroups $N_i$, $i=1,\ldots,n$, each
of order $p^3$, such that $G$ is the central
product of these factors.
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Sep 29
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We completed the sketch of the classification
of extraspecial $p$-groups.
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Oct 2
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We started on Chapter 8.
Group determinant.
Properties of characters.
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Oct 4
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We explained how to find the character table for any abelian
group, and for $A_4$. We defined representations (of groups
and categories). We defined permutation representations,
linear representations and $G$-modules.
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Oct 6
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We talked about changing language between
representations, actions, modules.
We defined permutation and linear representations,
and introduced the group algebra.
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Oct 9
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We showed that
$$\textrm{Hom}_{{\mathbb C}-\textrm{alg}}(\mathbb C[G],\mathbb A)\cong
\textrm{Hom}_{\textrm{Grp}}(G,\mathbb A^{\times}).$$
We started on the proof that if $G$ is a finite group,
then
$${\mathbb C}[G]\cong M_{d_1}({\mathbb C})\times \cdots\times M_{d_1}({\mathbb C}).$$
Se far we have shown that $\mathbb C[G]$ has finitely many
simple modules, all of which are finite dimensional.
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Oct 11
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We started the proof that if $A$
is a finite dimensional $\mathbb C$-algebra
and $V$ is a simple $A$-module, then the
associated representation $\rho\colon A\to End_{\mathbb C}(V)$
is surjective.
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Oct 13
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Schur's Lemma.
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Oct 16
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We discussed
Galois connections, and in particular
we determined the Galois connection between
a $k$-vector space $V$ and
the $k$-algebra $E:={\textrm End}_k(V)$
defined by the annihilation relation
$\rho = \{(\alpha,v)\in E\times V\;|\;\alpha(v)=0\}$.
We proved that the closed subsets of $V$ are exactly the subspaces
and the closed subsets of $E$ are the left ideals.
We derived the corollaries:
(0) the left ideal lattice of $E$ is dually isomorphic
to the subspace lattice of $V$, (1) $E$ is a simple $k$-algebra,
(2) $E$ has a unique isotype of simple module, namely $V$.
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Oct 18
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We proved that if if $V$ is a finite dimensional complex
vector space and $\mathbb A\subseteq \textrm{End}_{\mathbb C}(V)$
acts irreducibly on $V$, then every subspace of $V$ is closed in the
Galois connection defined by the relation
$\rho=\{(\alpha,v)\in \mathbb A\times V\;|\;\alpha(v)=0\}$.
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Oct 20
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We proved the Jacobson Density Theorem.
We defined the Jacobson radical and showed that
if $\mathbb A$ is a finite dimensional $\mathbb C$-algebra,
then $\mathbb A/J\cong \prod M_{d_i}(\mathbb C)$.
We explained why $J=\textrm{rad}(\mathbb A)$ has no complement
in the left ideal lattice of $\mathbb A$.
I stated that the next theorem to be proved, Maschke's Theorem,
which will show that if $G$ is a finite group and $k$ is a field
satisfying $\textrm{char}(k)\not\mid |G|$, then the left
ideal lattice of ${\mathbb A}$ is complemented.
Overall conclusion:
$\mathbb C[G]\cong \prod M_{d_i}(\mathbb C)$.
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Oct 23
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We proved Maschke's Theorem.
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Oct 25
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We started proving some of the basic properties of characters.
(Items (1)-(7) of the Oct 2 handout.)
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Oct 27
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We explained why the character table
of $G$ determines the set of normal ubgroups of $G$,
hence one can determine from the table if $G$ is simple
or solvable. We explained how to determine the character
table of $G/N$ from the character table of $G$ and from
the set $N$.
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Oct 30
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We justified (10)-(12), (14)-(17)
of the Oct 2 handout on properties of characters.
We also began a discussion of the
tensor product.
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Nov 1
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We justified (18)
of the Oct 2 handout on properties of characters.
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Nov 3
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We justified (19), (20)
of the Oct 2 handout on properties of characters.
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Nov 6
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We justified (21), (22)
of the Oct 2 handout on properties of characters.
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Nov 8
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We discussed row and column orthogonality.
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Nov 10
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We proved items (25)-(28).
We defined algebraic integers, and
stated a theorem characterizing them
as e-values of integer matrices
and also as those $z\in\mathbb C$
that generate subrings whose underlying
additive group is f.g.
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Nov 13
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We proved items (29)-(32).
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Nov 15
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We started these notes.
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Nov 17
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We finished the Nov 15 notes.
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Nov 27
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We began discussing the extension problem, $E: 1\to N\to G\to Q\to 1$.
We reviewed the left split case, the right split case,
and the unsplit case. The unsplit case reduced
to understanding a multiplication on
$Q\times N$
defined by $(q_1,n_1)\star(q_2,n_2) =
(q_1q_2,\gamma(q_1,q_2)\varphi(q_2)(n_1)n_2)$
where $\varphi\colon Q\to \textrm{Aut}(N)$ and $\gamma\colon Q\times Q\to N$
are functions.
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Nov 29
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We discussed how the
extension problem simplifies when the kernel, $N$, is abelian.
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Dec 1
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We defined $Z^2(Q,N), B^2(Q,N)$, and $H^2(Q,N)$.
We calculated these groups when $|Q|=2$.
We explained why, for example, $H^2(\mathbb Z_2,\mathbb Z)$
has size $2$ when $\mathbb Z_2$ acts trivially on $\mathbb Z$.
In this case, the two elements correspond to the two abelian extensions
of $\mathbb Z$ by a $2$-element group, namely the split extension
$0\to \mathbb Z\to \mathbb Z_2\times \mathbb Z\to \mathbb Z_2\to 0$ and
the nonsplit extension
$0\to \mathbb Z\stackrel{\times 2}{\to} \mathbb Z\to \mathbb Z_2\to 0$.
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Dec 4
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We defined $Z^n(Q,N), B^n(Q,N)$, and $H^n(Q,N)$,
and described some interpretations for
$H^n(Q,N)$ when $n=0, 1, 2, 3$.
We proved that is $|Q|=m$ and $N$ satisfies
$x^n=1$ for relatively prime $m$ and $n$, then
$H^2(Q,N)=0$. (The proof works for
$H^n(Q,N)=0$, too.)
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Dec 6
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We finished the proof of the Schur-Zassenhaus Theorem.
We defined "coprime action" and "cohomologically trivial".
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Dec 8
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Kevin Berg and Athena Sparks gave talks.
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Dec 11
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Mark Pullins and Ruofan Li gave talks.
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Dec 13
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Sarah Salmon and Steve Weinell gave talks.
FCQ's.
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