Date
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What we discussed/How we spent our time
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Aug 22
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Syllabus. Text.
We discussed that algebra is about computation. The word algebra is
derived from ``al-jabr'', which means ``restoring''.
It comes from the title of the first algebra book,
written by Al'Khwarizmi. We defined operation,
algebra,
partial algebra,
infinitary algebra, and
multisorted algebra, and gave examples ($+\colon \mathbb Z^2\to \mathbb Z$,
$\mathbb Z$ as a group,
$\mathbb R$ as a field, $[0,1]$ under the operation of lim sup,
and $\mathbb R^n$ as a vector space.) We noted that a category
is a 2-sorted partial algebra.
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Aug 24
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Today we discussed (informally) what an algebraic language is
and what a homomorphism is. We also discussed why, although
algebra is about calculation, algebraists spend time doing
other things, like creating algebraic models, understanding
the decomposition and reconstruction of algebraic models,
and classifying algebraic models. As an example
of the first of these we discussed the
algebraization of space.
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Aug 26
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Class canceled due to car accident.
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Aug 29
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We defined groups in general and the symmetric group, $S_X$, in particular.
We discussed the cycle representation of a permutation.
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Aug 31
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We discussed some examples of groups including $S_X$, $S_n$,
$\textrm{Aut}(\mathbb A)$ where $\mathbb A$ is a structure
(particular examples given where $\mathbb A$
was an undirected graph, a directed graph and a poset),
and the matrix groups
$\textrm{GL}(V)$ (or $\textrm{GL}_n(\mathbb R)$ when $V=\mathbb R^n$),
$\textrm{O}(V)$ (or $\textrm{O}_n(\mathbb R)$).
We pointed out why it is important to define `automorphism'
as `invertible morphism' rather than `1-1, onto morphism'.
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Sep 2
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We defined homomorphism, isomorphism and embedding.
We proved the Cayley Representation Theorem.
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Sep 7
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I circulated some inital definitions.
We refined the Cayley Representation Theorem by proving that
every group is the full automorphism group of a relational structure.
We defined the left and right regular representations.
We proved that a bijective homomorphism is an isomorphism,
and that an injective homomorphism is an embedding.
We started talking more generally about homomorphisms,
starting with this handout.
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Sep 9
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We discussed the image, coimage, kernel,
inclusion map, natural map and induced map associated to a function,
as well as the canonical factorization of a function.
We described the correspondence between coimages and kernels.
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Sep 12
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We gave examples of homomorphisms:
$\mathbb Z\to \mathbb R^{\times}\colon n\mapsto (-1)^n$ (so parity representation $\cong$ sign representation),
$\rho\colon
S_n\to \textrm{GL}_n(\mathbb R)\colon \sigma\mapsto \pi_{\sigma}$,
$\det\colon
\textrm{GL}_n(\mathbb R)\to \mathbb R^{\times}\colon M\mapsto \det(M)$,
$\textrm{sign}=\det\circ\rho\colon S_n\to \{\pm 1\}$.
We discussed images and kernels.
Our claims were:
there is a unique group structure on
$\textrm{im}(\varphi)$
that makes $\iota$ a homomorphism,
there is a unique group structure on $\textrm{coim}(\varphi)$
that makes $\nu$ a homomorphism,
these unique group structures make $\overline{\varphi}$ a homomorphism, and
$\textrm{im}(\varphi)$ and $\textrm{coim}(\varphi)$ satisfy universal properties.
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Sep 14
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We defined congruences, and noted the an equivalence relation
on $\mathbb Z$ is a congruence iff it is
`congruence modulo $n$' for some $n$.
We noted that a kernel is a congruence,
and conversely if $E$ is a congruence on $\mathbb A$ then
$\nu\colon \mathbb A\to \mathbb A/E$ is a homomorphism with kernel $E$.
We defined compatible relation and used this
notion to prove the First Isomorphism Theorem.
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Sep 16
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We began discussing left and right
invariant equivalence relations
on a group.
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Sep 19
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Professor Agnes Szendrei finished the handout on
invariant equivalence relations on groups.
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Sep 21
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Subgroup generation. Cyclic groups.
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Sep 23
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Subgroup lattices of cyclic groups.
Normal subgroup lattices are modular.
Subgroup lattice of $D_8$.
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Sep 26
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We reviewed previous material,
then discussed the universal property of products
in the context of topology.
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Sep 28
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We showed that the cartesian product of sets equipped
with the projection maps satisfies the universal property
of products.
We learned the meaning of the assertion ``the forgetful functor
from $\textrm{GRP}$ to $\textrm{SET}$ creates products''.
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Sep 30
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We defined products of groups, and argued that an identity
(= universally quantified equation)
holds in a product iff it holds in each of the factors.
We started on the proof of the characterization of products.
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Oct 3
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We finished the proof of the characterization of products.
We proved the Second Isomorphism Theorem.
We began discussing the Chinese Remainder Theorem for groups.
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Oct 5
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We finished discussing the Chinese Remainder Theorem.
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Oct 7
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We discussed the 3rd Iso Thm and the Correspondence Thm.
I circulated a
take-home test, due 10/17/16.
Solutions.
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Oct 10
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We started discussing representations of groups:
permutation representations, linear representations,
conjugation representation, actions, $G$-sets.
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Oct 12
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We continued discussing representations of groups:
homomorphisms, regular representation/free $G$-set,
orbit.
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Oct 14
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We discussed $\ker(\varphi)$ and $\textrm{im}(\varphi)$
for $\varphi$ a homomorphism with domain ${}_GG$.
We showed that a cyclic $G$-set ${}_GA$ is isomorphic to
${}_G(G/H)$ for $H$ equal to the stabilizer of some $a\in A$.
Orbit-Stabilizer Theorem.
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Oct 17
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We showed that there is a homomorphism
$\varphi\colon {}_G(G/H)\to {}_G(G/K)\colon H\mapsto gK$ iff $H\leq gKg^{-1}$,
and that ${}_G(G/H)$ is isomorphic to ${}_(G/K)$ iff
$H$ and $K$ are conjugate.
We defined coproduct, and explained why every
$G$-set is the coproduct of its cyclic subalgebras.
We used this to describe all objects and morphisms in the category
of $G$-sets for a fixed $G$.
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Oct 19
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We explained why, if ${}_GA$ and ${}_GB$ are $G$-sets,
then there is a natural way to define a $G$-action on the set of functions
$B^A$. In fact, if $A$ and $B$ are disjoint,
then there is a natural way to define a $G$-action on the
set-theoretical universe over the set of atoms $A\bigcup B$.
Class equation.
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Oct 21
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There is a group $G$ where $n$ divides $|G|$, but $G$ has
no subgroup of order $n$. However, if $G$ is solvable and $n$ is a Hall
divisor, then $G$ must have a subgroup of size $n$.
Cauchy's Theorem. The First Sylow Theorem.
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Oct 24
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We finished the proofs of the Sylow Theorems.
We defined restricted action.
We introduced a version of the class equation
for any action.
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Oct 26
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We showed that any group of order $pq, p^2q, pq^2$, $p
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Oct 28
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We finished discussing semidirect products, including the statements
of the Schur-Zassenhaus Theorem and the Burnside Normal Complement Theorem.
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Oct 31
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We discussed the classification of small groups. Our discussion
involved: the elementary divisor form and invariant factor form
for a finite abelian group, the fact that $p$-group
has a nontrivial center, then fact that $G/Z(G)$ cyclic implies
$G$ abelian, the structure of groups of order $p$, $p^2$, or $pq$,
and the characterization of those $n$ for which there
is only one isotype of group of order $n$.
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Nov 2
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We finished the proof of the characterization of those $n$
for which all groups of order $n$
are cyclic, and stated the
characterization of those $n$ for which all groups of order $n$
are abelian. We identified all groups up to order $15$.
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Nov 4
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We began discussing groups of order $p^3$.
Along the way we proved that maximal subgroups of $p$-groups
are normal, and the $p$-groups have the normalizer property.
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Nov 7
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We finished discussing groups of order $p^3$.
Along the way we introduced the definitions of the holomorph and
of an affine group.
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Nov 9
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The commutator. Solvable groups. Nilpotent groups.
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Nov 11
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Universal properties,
free groups, and presentations.
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Nov 14
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More about
free groups.
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Nov 16
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Subnormal subgroups,
composition series, Jordan-Hölder Theorem,
simple groups.
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Nov 18
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We introduced rings as algebraic models of
endomorphism structures of abelian groups.
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Nov 28
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We talked about ideals.
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Nov 30
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We sketched the construction of the field of fractions
of an integral domain.
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Dec 2
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We talked about homomorphisms into fields, and noted that
the kernel of such a homomorphism is prime. Thus,
$I\lhd R$ is prime iff $R/I$ is a subring of a field
(an integral domain),
while $I\lhd R$ is maximal iff $R/I$ is a field.
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Dec 5
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Polynomial rings.
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Dec 7
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Unique factorization.
Practice problems.
We also explained why $\mathbb Z[i]$ is a Euclidean domain,
why $\mathbb Z[\sqrt{-2}]$ is a Euclidean domain, but
$\mathbb Z[\sqrt{-3}]$ is not a Euclidean domain (since it is not a UFD:
$(1+\sqrt{-3})(1-\sqrt{-3})=4=2\cdot 2$ are two inequivalent
factorizations into irreducibles). We defined ACC and DCC for an ordered set,
and ``Noetherian ring'' and ``Artinian ring''.
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Dec 9
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We discussed practice problems and
I distributed an exam.
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