Date
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What we discussed/How we spent our time
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Aug 24
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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 described three examples
of algebraic structures: the group $\mathbb Z$, the vector space
$\mathbb R^n$, and a 2-sorted algebra modeling the
stack data structure.
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Aug 26
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We discussed this handout.
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Aug 28
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We reviewed the definitions of ``homomorphism'' and ``isomorphism''.
(We also discussed ``isomorphic'' and ``$\cong$''.) We determined which
of the following algebras are isomorphic:
$
\mathbb A = \langle \{0,1\}; \star\rangle,
\mathbb B = \langle \{0,1\}; \star\rangle,
\mathbb C = \langle \{0,1\}; \star\rangle$
where
$$
\begin{array}{|r||c|c|}
\hline
\star^{\mathbb A}& 0 & 1\\
\hline
\hline
0& 0 & 0\\
\hline
1& 0 & 1\\
\hline
\end{array}
\quad
\begin{array}{|r||c|c|}
\hline
\star^{\mathbb B}& 0 & 1\\
\hline
\hline
0& 1 & 0\\
\hline
1& 0 & 0\\
\hline
\end{array}
\quad
\begin{array}{|r||c|c|}
\hline
\star^{\mathbb C}& 0 & 1\\
\hline
\hline
0& 0 & 1\\
\hline
1& 1 & 1\\
\hline
\end{array}
$$
We then defined ``groups'' (and ``abelian groups'')
and classified all groups of size $0, 1$ and $2$.
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Aug 31
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We explained why the inverse and identity tables
of a group can be determined
from the multiplication table. We showed that the
multiplication table of a group is a Latin square.
We found that there is one $3$-element group up to isomorphism
and two $4$-element groups up to isomorphism.
Quiz 1.
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Sep 2
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We reviewed some easily-proved properties of group tables.
We reviewed our list of small groups. We introduced the
groups $\mathbb Z_n$, and explained that $\mathbb Z_n$
is isomorphic to the group of rotations of the circle through
angles that are multiples of $\frac{2\pi}{n}$.
We defined the subgroup $\langle X\rangle\leq G$ generated
by a subset $X\subseteq G$,
and defined
cyclic groups.
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Sep 4
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We showed that the subgroups of a group form a lattice
under the inclusion order, and that the lattice operations
are $H\wedge K = H\cap K$ and $H\vee K = \langle H\cup K\rangle$.
We determined the subgroup lattices of the 4-element groups.
We defined the order of an element
of a group. We proved that if $a\in G$ has infinite order,
then $\langle a\rangle$ is isomorphic to $\mathbb Z$.
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Sep 9
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Today we deviated from the book to try to sketch an answer to the question
``What is algebra for?'' We discussed the algebraization process,
and focused on the question ``What are the laws of functional composition?''
This led to us defining semigroups and monoids, and to proving
the Cayley Representation Theorem for monoids. This theorem asserts
that every monoid is isomorphic to a monoid of functions.
We derived from this that there is no universally quantified law
of functional composition other than the consequences of the
associative law and the laws for the identity element.
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Sep 11
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We discussed some mathematical objects and their
algebraic models: functions under composition (semigroups),
1-1 functions under composition (left cancellative semigroups),
onto functions under composition (right cancellative semigroups),
1-1, onto functions (groups), logical propositions (Boolean algebras).
We then returned to our study of groups. We defined permutations,
described the cycle representation of a permutation, explained how to
multiply and invert permutations written in cycle form,
defined the symmetric group on a set,
and started computing the multiplication table for $S_3$.
(We got far enough through this to see that $S_3$ is not abelian.)
We observed that $S_n$ has $n!$ elements.
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Sep 14
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We discussed the Cayley Representation Theorems for semigroups
and groups. Then we discussed examples of groups of permutations,
namely: the group of rotations of Euclidean space, the group
of rotations of an $n$-gon, the dihedral groups, the groups of rotations
of the Platonic solids, and automorphism groups of algebraic structures.
Quiz 2.
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Sep 16
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Today we discussed the name, notation, elements, order, manner of composition,
and presentation of the finite groups $S_n$, $C_n$, $D_n$, $Q_{4n}$.
We gave a table of all small groups up to size $12$.
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Sep 18
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Please read Section 4.1.
We discussed how to represent any finite group as a group
of orthogonal matrices in dimension $|G|$, or as a group
of rotation matrices in dimension $|G|+2$. ($M$ is a rotation
matrix if $M$ is orthogonal and has determinant $1$.)
We discussed
the rotation groups of the Platonic solids.
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Sep 21
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Please read Sections 2.4-2.7.
We reviewed all material up to this point, then discussed
this handout that describes
the canonical factorization of a function.
We then began to rework the handout under the assumption
that $f\colon A\to B$ is a homomorphism of groups. We proved that
$\textrm{im}(f)$ is a subgroup of $B$, then stated
without proof that
$\textrm{coim}(f)$ is a quotient group of $A$ and that
$\nu, \overline{f}, \iota$ are homomorphisms.
Quiz 3.
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Sep 23
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We proved that there is a unique way to define
a group structure on the image of a homomorphism
so that the inclusion map is a homomorphism.
We proved that there is a unique way to define
a group structure on the coimage of a homomorphism
so that the natural map is a homomorphism.
For these group structures, the induced map
is an isomorphism.
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Sep 25
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We reviewed the last HW assignment, then defined
cosets of subgroups
and indicated their properties.
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Sep 28
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We discussed properties of cosets and proved that
$|G| = [G:H] |H|$.
Quiz 4.
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Sep 30
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We discussed normality.
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Oct 2
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We discussed quotient groups and the First Isomorphism Theorem.
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Oct 5
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We discussed conjugacy and its connection to normality.
We noted that, if $a\in G$, then
$c_a(x) = axa^{-1}$ is an automorphism
of $G$ and the function
$G\to \textrm{Aut}(G)\colon a\mapsto c_a$ is a homomorphism.
We explained why permutations in $S_n$ are conjugate iff
they have the same cycle type.
Quiz 5.
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Oct 7
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We proved the Second Isomorphism Theorem.
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Oct 9
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We proved the Third Isomorphism Theorem and half
of the Correspondence Theorem. (In one example
we explained why there are a lot of subgroups
between $SL_n(\mathbb R)$ and $GL_n(\mathbb R)$,
all normal.)
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Oct 12
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We completed the proof of the
Correspondence Theorem.
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Oct 14
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We discussed the review sheet.
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Oct 16
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Midterm.
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Oct 19
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We defined products of groups. We defined projection homomorphisms.
We explained how homomorphisms into a product are determined
coordinatewise, and formulated the Universal Property of Products.
We stated a theorem characterizing products in terms of
complementary normal subgroups.
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Oct 21
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We proved the theorem characterizing products of groups.
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Oct 23
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We talked about defining a kind of metric
a group, namely $d(a,b) = \langle a^{-1}b\rangle$.
We explained the geometric content
of the Chinese Remainder Theorem for groups in this language.
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Oct 26
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We discussed factoring cyclic groups, and stated the
Fundamental Theorem of Finite Abelian Groups.
The statement included the claim that two finite abelian
groups are isomorphic iff their elementary divisors are equal.
Quiz 6.
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Oct 28
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We introduced the semidirect product construction. (Section 3.2.)
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Oct 30
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We reviewed the semidirect product construction and considered some examples.
The examples included ordinary product groups, dihedral groups,
generalized dihedral groups, infinite dihedral groups, and affine
groups.
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Nov 2
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We introduced $G$-sets, actions and representations.
Among our examples was the left regular $G$-set/action/representation,
written as an algebra as $\langle G;G\rangle$. We explained why,
if $x_0\in X$, there is a unique
homomorphism $h\colon \langle G;G\rangle\to \langle X;G\rangle$
such that $h(e)=x_0$. Namely it is the homomorphism
described by $h(g) = g(x_0)$.
Quiz 7.
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Nov 4
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We explained why $\langle G; G\rangle$ is free over
the basis $\{e\}$. We defined orbit, stabilizer and
proved the Orbit-Stabilizer Theorem
($|G|=|\textrm{Stab}(x_0)|\cdot|{\mathcal O}(x_0)|$).
We determined the sizes of the orbits and stabilizers
of the vertices, edges and faces of a cube
under the action of its rotation group.
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Nov 6
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We reviewed the Orbit-Stabilizer Theorem, and
proved that the size of the conjugacy class of $g\in G$
is $[G\colon C_G(g)]$. We then discussed Not-Burnside's Lemma,
and applied it to count the number of
inequivalent ways to arrange distinct numbers
on the face of a cube.
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Nov 9
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Everybody knows that the total number of 4-legged cows in a field
is #legs/4.
We showed that Not-Burnside's Lemma could be used
to compute the number of cows in a field
in a situation where some cows have three legs and some
cows have four legs. Then we
worked on practice problems.
Quiz 8.
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Nov 11
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We proved the Cauchy-Frobenius Formula (from Not-Burnside's Lemma),
and used it to
solve Problem 7 on the worksheet from the last lecture.
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Nov 13
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We examined the subgroup lattices of groups
of order $\leq 8$. We proved Cauchy's Theorem.
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Nov 16
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We introduced the Class Equation and the center ($Z(G)$),
and proved that a group of order $p^k$ has a nontrivial center.
Quiz 9.
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Nov 18
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We proved that: (i) $Z(G)\lhd G$, (ii) $G=Z(G)$ iff $G$ is abelian, and
(iii) if $G/Z(G)$ is cyclic, then $G$ is abelian. We derived that
a group of order $p^2$, $p$ prime, is abelian, and hence isomorphic
to $\mathbb Z_{p^2}$ or to $\mathbb Z_p\times \mathbb Z_{p}$.
We introduced the Sylow Theorems.
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Nov 20
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We discussed the Sylow Theorems (statement and importance).
We showed how to use the theorems to classify groups of order $pq$,
where $p \lt q$ are prime. We introduced the notion of exact sequence
and of simple group. We described some results on the ``range problem''
which is the problem of determining the structure of
all groups of a given order. We noted that there are no
simple groups of order $p^k$, $k>1$; $pq$, $p^2q$, or $pqr$.
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Nov 30
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Lemma. If a $p$-group $H$ acts on a set finite set $X$, then
$|X|\equiv |\textrm{Fix}_H(X)|\pmod{p}$.
We used this to prove the First and Second Sylow Theorems.
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Dec 2
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We proved the Third Sylow Theorem and applied it in some problems.
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Dec 4
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We talked about the statement of the
Classification of Finite Simple Groups.
We began discussing the alternating groups.
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Dec 7
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I circulated a review sheet.
We showed that the sign map is a homomorphism,
$\textrm{sgn}\colon S_n\to \{+1,-1\}$, that is surjective when $n>1$.
The kernel is $A_n$, so $A_n\lhd S_n$ and $[S_n:A_n]=2$ when $n>1$.
We explained why the Cauchy number of a permutation $\sigma$
is the smallest $k$ such that $\sigma$ can be written
as a product of $k$ transpositions.
Quiz 10.
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Dec 9
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We showed that
if $[S:A]=2$, $a\in A$, and
-
$[S:C_{S}(a)]\not\subseteq A$, then
the $S$-conjugacy class of $a$ equals the $A$-conjugacy
class of $a$, while
-
$[S:C_{S}(a)]\subseteq A$, then
the $S$-conjugacy class of $a$ splits into two
$A$-conjugacy classes.
We used this to show that the conjugacy class sizes in $A_5$
are: $1, 12, 12, 15, 20$. Using this we showed that $A_5$ is simple.
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