Date
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What we discussed/How we spent our time
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Aug 23
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Prerequisites.
Syllabus. Text.
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Aug 25
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We reviewed (and gave examples where appropriate) the following:
(i) given a function $f\colon A\to B$:
$\textrm{dom}(f)$, $\textrm{cod}(f)$, $\textrm{im}(f)$,
$\textrm{coim}(f)$,
$\textrm{ker}(f)$, inclusion map, natural map, induced map,
and the canonical factorization of $f$.
(ii) The definition of `algebraic structure'.
(iii) The purpose of fixing an algebraic language.
(iv) The definition of `homomorphism'.
(v) The definition of `subalgebra' and `quotient algebra'.
(vi) The definition of `group' and `abelian group'.
(vii) given a group homomorphism $f\colon \mathbb A\to \mathbb B$:
$\textrm{dom}(f)$, $\textrm{cod}(f)$, $\textrm{im}(f)$,
$\textrm{coim}(f)$,
$\textrm{ker}(f)$,
$\textrm{Ker}(f)$.
(viii) The definition of `exact sequence' of algebras
that are expansions of groups.
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Aug 27
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We talked about:
(i) products, coproducts, biproducts.
(ii) Retractions and sections.
Exact sequences split on the left or split on the right.
We explained why an exact sequence of groups that is split
on the left must also be split on the right.
(iii) Projective and injective modules.
We explained why free modules are projective.
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Aug 30
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We talked about:
(i) the structure theorem for finitely generated
modules over a PID applied to
finitely generated abelian groups.
In particular, every finitely generated abelian group
is a direct sum of cyclic groups. Also, every
finitely generated torsion free abelian group is free.
(ii)
the set of torsion elements of an abelian group
is a subgroup $A_T$. This fact also holds for nilpotent groups,
but fails for some solvable groups like
$D_{\infty}$ or the
triangle group $(3,3,3)$.
(iii)
if $A$ is an $n$-generated abelian group and $A_T$ is the
set of torsion elements of $A$, then
(a) $A_T$ is an $n$-generated torsion group.
(b) $A/A_T$ is torsion-free.
(c) the exact sequence
$$(E):\;\;0\to A_T\to A\to A/A_T\to 0$$
splits.
(d) $A_T$ is finite and a direct sum of cyclic groups.
(e) $A/A_T$ is free on at most $n$ generators.
(iv)
We introduced the Prufer $p$-groups ($p$-quasicyclic groups)
and explained why the exact sequence
$$(E):\;\;0\to \oplus_p \mathbb Z_{p^{\infty}}\to \prod_p \mathbb Z_{p^{\infty}}
\to \mathbb Q^{\aleph_0}\to 0$$
fails all parts of (c), (d), and (e) from above.
(v)
We proved that if $P$ is projective, then
any exact sequence $0\to A\to B\to P\to 0$
splits.
We started a proof of the converse, by showing that
if $0\to K\to F\to P\to 0$ splits when $F$ is free,
then $F=K\oplus P$, so $P$ is a direct summand of a free
abelian group. To complete the converse we need
to show that a direct summand of a free module is projective.
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Sep 1
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We expanded the statement of the characterization
of projective abelian groups so that it reads: TFAE
(i) $P$ is projective.
(ii) Every SES $0\to A\to B\to P\to 0$ splits.
(iii) Some SES $0\to K\to F\to P\to 0$ splits, where $F$ is free.
(iv) Some SES $0\to K\to F\to P\to 0$ splits, where $F$ is projective.
(v) $P$ is a direct summand of a free abelian group.
We discussed the dualization of this proof, noting that
item (iii) does not dualize well.
Instead, the proof should be that TFAE
(i) $I$ is injective.
(ii) Every SES $0\to I\to B\to C\to 0$ splits.
(iv) Some SES $0\to I\to J\to K\to 0$ splits, where $J$ is injective.
(v) $I$ is a direct summand of an injective group.
The proof of this dual theorem
requires the knowledge that every abelian group
is embeddable in an injective abelian group
(to go from (i) to (iv)).
We sketched the reason for this:
if $A^{\partial}=\textrm{Hom}(A,\mathbb T)$
for $\mathbb T$ = circle group $\cong \mathbb R/\mathbb Z$,
then $A^{\partial}$ is called the
character group (or dual group) of $A$.
$A$ is embeddable in its double dual
by $a\mapsto (\chi\mapsto \chi(a))$.
The double dual group is a subgroup of
$\mathbb T^{A^{\partial}}$, which is injective
since $\mathbb T$ is injective and
the class of injective groups is closed
under the formation of products. Thus
$A\to (A^{\partial})^{\partial}\to\mathbb T^{A^{\partial}}$
is an embedding of $A$ into an injective abelian group.
We concluded by remarking that the class of
injective abelian groups coincides with the class
groups isomorphic to direct sums of
copies of the groups $\mathbb Q$ and $\mathbb Z_{p^{\infty}}$
for $p$ prime.
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Sep 3
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We talked about:
(i) the proof that a
subgroup of a free abelian group is free abelian.
(Hence an abelian group is projective if and only if it is free.)
(ii)
the fact that an injective group is divisible (= principally injective).
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Sep 8
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Injective versus divisible abelian groups:
(i) proof that an abelian group is divisible if and only if
it is injective.
(ii) Remarks: (some stated, some proved)
(a) If $\gcd(m,n)=1$ and $a\in A$ is divisible by
both $m$ and by $n$, then $a$ is divisible by $mn$.
(b) The class of injective groups is closed under the formation
of products and retractions.
(c) The class of divisible groups is closed under the formation
of direct sums and quotients.
(d) Every abelian group is embeddable in a divisible group.
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Sep 10
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We explained why:
(i) the torsion subgroup and torsion-free quotient
of a divisible group $D$ are divisible, and $D\cong D_T\oplus D/D_T$.
(ii) A torsion-free divisible group is isomorphic to a direct sum of copies
of $\mathbb Q$.
(iii) A torsion group is a direct sum if its $p$-primary components.
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Sep 13
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We completed the proof of the structure theorem for divisible groups.
(New terminology: essential embedding, socle, semisimple module.)
We defined the Steinitz numbers and explained how they classify the
intermediate fields $\mathbb F_p\leq \mathbb E\leq \overline{\mathbb F}_p$.
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Sep 15
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Let $A$ be an abelian group and let $a$ be an element of $A$.
We defined the $p$-height of $a$, the
characteristic of $a$, and the type of $a$.
We started the proof of Baer's Theorem, which states that
the nonzero elements of a torsion-free abelian group
of rank 1 have the same type, and this type
determines the isomorphism type of the group.
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Sep 17
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We completed the classification of t.f. abelian groups
of rank $1$. (All nonzero elements have the same type, and the type
determines the isomorphism type of the group.)
We examined an indecomposable, t.f., abelian group of rank $2$,
which has the properties that (i) it has elements of different types,
(ii) its only automorphisms
are of the form $\alpha(x) = \pm x$, and (iii) its
only endomorphisms are of the form $\varepsilon(x) = n x$, $n\in\mathbb Z$.
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Sep 20
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We discussed
torsion abelian groups.
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Sep 22
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We defined the commutator following
this handout, and discussed
properties of commutators of elements
and commutators of normal subgroups.
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Sep 24
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We continued discussing the commutator following
this handout.
We applied the commutator to classify groups whose
normal subgroup lattice has height 2 and size at least 5.
We noted that $[G,G]$ is verbal, that
$Z(G)=(0:G)$ is marginal, and that it is a general fact that
verbal subgroups are fully invariant
and marginal subgroups are characteristic.
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Sep 27
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We discussed
(1) Derived series, central series.
(2) The groups $\textrm{GL}(n,\mathbb F), \textrm{UT}(n,\mathbb F), \textrm{U}(n,\mathbb F)$.
We explained why the unipotent group $\textrm{U}(n,\mathbb F)$ is nilpotent for every $n$ and $\mathbb F$, and why the group
$\textrm{UT}(n,\mathbb F)$ is solvable for every $n$ and $\mathbb F$.
We explained why, for every $n$ and finite field $\mathbb F$ of characteristic $p$, the group $\textrm{U}(n,\mathbb F)$ is a $p$-group
(in fact a Sylow $p$-subgroup of $\textrm{GL}(n,\mathbb F)$).
We also explained why, for any finite field $\mathbb F$ of characteristic $p$, any given finite $p$-group can be embedded in $\textrm{U}(n,\mathbb F)$ for sufficiently large $n$.
(3) Commutator collection.
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Sep 29
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We discussed the classification of $p$-groups.
After some generalities, we described the groups
of order $p^3$ ($D_4, Q_8$, and the Sylow $p$-subgroups
of $\textrm{GL}(3,p)$ and $\textrm{AGL}(1,\mathbb Z_{p^2})$).
We then defined extraspecial $p$-groups
and made some observations.
(1) The normal subgroup lattice of a nonabelian
group $P$ of order $p^3$ has one element $\{1\}$ at height $0$,
one element $Z(G)$ at height $1$,
and $p+1$ elements at height $2$, and
one element $P$ at height $3$.
(2) If $1\neq N\lhd G$ and $G$ is nilpotent,
then $N\cap Z(G)\neq 1$.
(3) The number of abelian groups of order $p^k$
is given by the partition function, which grows
roughly at the rate $e^{\Theta(\sqrt{k})}$.
(4) (Higman-Sims) The number of groups of order $p^k$
is $p^{\frac{2}{27}k^3+O(n^{\frac{8}{3}})}$.
(5) In an extraspecial $p$-group $G$,
$Z(G)=G'=\Phi(G)$ and
$G/Z$ is elementary abelian.
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Oct 1
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We discussed Philip Hall's 1940 paper
in which he introduced isoclinism of groups.
We saw that in an extraspecial $p$-group $G$,
the Hall structure Hall($G,[x,y]$)
``is'' a vector space over $\mathbb F_p$
equipped with a nondegenerate, alternating,
bilinear form.
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Oct 4
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We showed that simplectic
spaces over $\mathbb F$ are determined up to isomorphism
by their dimension. We outlined how this could be
used to help classify extraspecial $p$-groups.
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Oct 6
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We discussed central products
and outlined how they are used to describe
extraspecial $p$-groups.
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Oct 8
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We defined nongenerators, the Frattini subgroup,
and the Fitting subgroup. We explained why
(1) $\textrm{Frat}(G)$ and $\textrm{Fit}(G)$ are characteristic subgroups.
(2) The Frattini subgroup of $G$
is the set of nongenerators of $G$.
(3) The Frattini subgroup of a finite group is nilpotent.
(4) Fitting's Theorem: the join of two nilpotent normal subgroups
is a nilpotent normal subgroup.
(5) If $G$ is a finite solvable group and $F$ is its Fitting subgroup, then
$C_G(F)=(0:F)_G$ is equal to $Z(F)$.
We discussed how the Fitting subgroup ``controls/determines''
the structure of a finite solvable group. Namely, if $\textrm{Fit}(G)=F$
and $\Gamma\colon G\to \textrm{Aut}(F)\colon g\mapsto \gamma_g$
has kernel $(0:F)_G = Z(F)$ and image $I\leq \textrm{Aut}(F)$,
then there is an exact sequence
$$
0\to Z(F)\to G\to I\to 0$$
where $I$ is a solvable subgroup of
$\textrm{Aut}(F)$ containing $\textrm{Inn}(F)$. This shows that, given finite nilpotent $F$,
there are finitely many finite, solvable $G$ such that $F=\textrm{Fit}(G)$.
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Oct 11
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We began discussing the definition and basic properties
of the transfer (Section 10.1 only).
(1) We discussed the transfer (Verlagerung) $V_{G\to H}(x)$ of
$G$
into an abelian subgroup $H\leq G$
of finite index
and transfer $V_{G\to K}(x)$
of $G$ into an arbitrary abelian group $K$.
(2) If $R=(r_1,\ldots,r_n)$ and $S=(s_1,\ldots,s_n)$ are ordered
transversals for $H$ in $G$, and $Hr_i=Hs_i$ holds
for all $i$, then define $(R|S)= \overline{\prod r_is_i^{-1}}$.
We showed
(i) $(R|S)^{-1}=(S|R)$.
(ii) $(R|S)(S|T)=(R|T)$.
(iii) $(\forall g\in G)$ $(Rg|Sg)=(R|S)$.
(iv) $(\forall R, S)(\forall g\in G)$ $(Rg|R)=(Sg|S)$.
(v) $V(g)=(Rg|R)$.
(vi) $V$ is a group homomorphism.
(3) Formula for $V$:
$V(g)=\overline{(s_1g^{k_1}s_1^{-1})\cdots (s_ug^{k_u}s_u^{-1})}$.
(4) If $H\leq Z(G)$, $[G:H]=n$, then $V_{G\to H}(x)=x^n$.
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Oct 13
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We continued discussing basic properties
of the transfer.
We noted that if $[G:H]=n$ and $H$
has a normal complement $T$, then
$V_{G\to H}(x)\equiv x^n\pmod{[H,H]}$ for $x\in H$ and
$V_{G\to H}(x)\equiv 1\pmod{[H,H]}$ for $x\in T$.
We began discussing whether the converse is true,
and stated the Burnside Normal $p$-Complement Theorem.
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Oct 15
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We proved:
(1) If $P\in \textrm{Syl}_p(G)$, then $N_G(P)$ controls
fusion in $C_G(P)$.
(2) If $P\in \textrm{Syl}_p(G)$ and $N_G(P)=C_G(P)$,
then $P$ is abelian and no $x,y\in P$, $x\neq y$,
are conjugate in $G$.
(3) The Burnside Normal $p$-Complement Theorem.
(4) B's N/C Theorem leads to a quick classification of
simple groups of order $60$.
We stated:
(5) (Thm 10.1.6 of Robinson) If
$P\in \textrm{Syl}_p(G)$ is abelian and $N=N_G(P)$, then
$\textrm{im}(V_{G\to P})= C_P(N)$,
$\textrm{ker}(V_{G\to P})|_P= [P,N]$,
and $P\cong C_P(N)\times [P,N]$.
(6) (Thm 5.20 of Isaacs)
If $P\in \textrm{Syl}_p(G)$, then
$\textrm{ker}(V_{G\to P_{\tt ab}})= {\bf A}^p(G)$.
(7) The classification of finite groups with cyclic subgroups.
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Oct 18
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We proved Item (7) from October 15: a group with
cyclic Sylow subgroups has the form
$\mathbb Z_m\rtimes \mathbb Z_n$ for some $m, n$
satisfying $\gcd(m,n)=1$.
(The theorem of Hölder-Burnside-Zassenhaus.)
As a corollary, we obtain a characterization
of those numbers
$n$ such that the only group of order $n$
is the cyclic group $\mathbb Z_n$.
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Oct 20
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We discussed
(1) Theorem. (Schur) If $[G:Z(G)]<\infty$, then $|[G,G]|<\infty$.
(2) Theorem. If $G$ is f.g. and $[G:H]<\infty$,
then $H$ is f.g.
(3) Theorem. If $\mathbb A$ is a f.g. algebraic structure
with finitely many basic operations
and $\theta$
is a congruence of finite index on $\mathbb A$,
then $\theta$ is finitely generated.
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Oct 22
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We discussed the group determinant.
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Oct 25
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We finished the group determinant
slides, and introduced some of the basic
definitions concerning the representation of groups.
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Oct 27
|
We discussed the main
ideas for the representation theory
of finite groups over the complex numbers.
We discussed the isomorphism types for $\mathbb Q[\mathbb Z_p]$,
$\mathbb R[Q_8]$, and $\mathbb C[Q_8]$.
We defined and discussed $k$-algebras in the case
where $k$ is a field. (E.g., $k, M_n(k), k[G]$.)
We noted that there are $2^{2^{\aleph_0}}$-many
distinct ways to view $\mathbb C$ as a
$\mathbb C$-algebra, but the most natural
one arises from the identity embedding
$\mathbb C\hookrightarrow\mathbb C$.
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Oct 29
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We proved that
$\textrm{Hom}_{k-\textrm{alg}}(k[G],\mathbb A)\cong
\textrm{Hom}_{\textrm{Grp}}(G,\mathbb A^{\times})$.
We proved Maschke's Theorem.
We discussed semisimple modules and algebras.
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Nov 1
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We discussed the structure of
the $\mathbb C$-algebra $\mathbb C[G]$ when
$G$ finite, along with some properties
of its category of modules.
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Nov 3
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We completed the
semisimplicity handout.
We discussed the representations
of $S_3$.
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Nov 5
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We started discussing
properties of characters.
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Nov 8
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We computed $K_{\chi_i}$ and $Z_{\chi_i}$
for $S_3$ and $Q_8$,
for all $i$, and showed how to recover the normal
subgroup lattice of the group from this information.
We saw how to determine the indices between normal subgroups.
We started explaining how to determine solvability
of a group from the character table.
For this we showed:
(1) If $G$ is a finite subdirect product of
simple groups, then it is a full product
of some subset of the factors.
(2) If $G$ is a finite characteristically simple
group, then $G$ is a power of a finite simple group.
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Nov 10
|
We compared the character tables for
$S_3$ and $S_4$, computing $K_{\chi}$
for all $\chi\in\textrm{Irr}(G)$ for both groups.
We completed part of the Monday discussion by explaining why:
(3) A minimal normal subgroup of a finite
group is characteristically simple.
(4) A subset of a finite group $G$ is a normal subgroup
iff it is the intersection of kernels of irreducible
characters.
(5) A finite group is solvable iff it has a chief series
whose factors have prime-power order.
(6) A group of order $p_1^{e_1}\cdots p_r^{e_r}$
is nilpotent iff it has normal subgroups of order
$p_i^{e_i}$ for all $i$.
We defined the regular representation of a group.
We defined inflation of a representation or character.
We explained the relationship between
the irreducible characters of $G/N$ and the irreducible
characters $\chi$ of $G$ satisfying $K_{\chi}\supseteq N$.
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Nov 12
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We discussed the relationship
between the character tables of $Q_8$ and $Q_8/Z(Q_8)$.
We discussed Theorems (12) and (13)
of this handout.
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Nov 15
|
We finished discussing Page 1
of this handout
(the part that involves Dimension and Kernel and Center).
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Nov 17
|
We worked through
these slides
up to, but not including, the determination
of $\chi_{U^*}$ from $\chi_U$.
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Nov 19
|
We completed the last two pages of
these slides
and then completed these slides.
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Nov 29
|
(Orthogonality relations)
We continued the discussion of
this handout
following these slides.
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Dec 1
|
We completed the orthogonality slides,
as well as most of the
integrality slides.
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Dec 3
|
André Davis gve a lecture about
formal group laws. This included
(1) The definition.
(2) Many examples of formal group laws.
(3) A description of the functor that assigns to a commutative ring
its set of formal group laws.
(4) The statement that the
functor is representable by the Lazard Ring $L$.
We finished the
integrality slides,
and thereby completed this handout.
At the end of the lecture, we noted that
(i) if $\rho\colon G\to \textrm{GL}(m,\mathbb C)$
is an irreducible representation
of $G$ and $\varepsilon\colon G\to G$ is an automorphism,
then $\rho\circ \varepsilon$ is an irreducible representation
of $G$, and (ii) if
$\varphi\colon \textrm{GL}(m,\mathbb C)\to
\textrm{GL}(n,\mathbb C)$ is a surjective
homomorphism, then $\varphi\circ \rho$
is an irreducible representation.
(Most important special cases of (ii): $\varphi = \det$
or $\varphi$ is induced by an automorphism of $\mathbb C$.)
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Dec 6
|
We discussed examples and applications of character tables
following these slides.
This included
(1) Character tables of abelian groups.
(2) Character tables of product groups.
(3) Character tables of $Q_8$ and $D_4$.
(4) Burnside's $p^aq^b$ Theorem.
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Dec 8
|
We completed these slides.
This included
(1) Character table of $A_5$.
(2) Proof that if a character $\psi$ satisfies $\psi(1)=p^k$
and $P\in \textrm{Syl}_p(G)$, then $\psi$
vanishes on $Z(P)-\{1\}$.
(3) Proof that if a character vanishes on $H-\{1\}$,
then it is an integer multiple of the regular character on $H$.
(4) Proof that if $G$ is a nonabelian simple group
and some character $\psi$ satisfies $\psi(1)=p$,
then the Sylow $p$-subgroups of $G$ are cyclic of order $p$.
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