Lecture Outlines
We will cover a wide variety of materials during lecture and discussion sections, so your constant attendance is important. To help you in organizing your study materials, the list below gives an overview of the basic concepts covered during a given lecture period.
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We started class today by discussing some of the highlights of the syllabus; YOU ARE RESPONSIBLE FOR READING AND UNDERSTANDING EVERYTHING ON THE SYLLABUS.
Afterwards, we started our pursuit of algebra by introducing some number theoretic basics. We listed a few of the axioms of $\mathbb{Z}$, taking particular care to discuss the Well-Ordering Principle in particular. It says
We then went on to prove one of the fundamental results in elementary number theory: the Division Algorithm.Axiom (Well-Ordering Principle)
If $S \subseteq \mathbb{Z}$ is nonempty and there exists some $z \in \mathbb{Z}$ so that for all $s \in S$ we have $z \leq s$, then there exists some $s_0 \in S$ so that for all $s \in S$ we have $s_0 \leq s$..
We defined the notion of divisibility in $\mathbb{Z}$, and we observed that the division algorithm gives us a way to understand divisibility in a slightly more refined way; instead of having only a "yes" or "no" answer to whether $d$ divides $a$, the remainder term in the division algorithm gives us a more precise way to measure precisely how divisibility fails.Theorem (Division Algorithm)
Suppose that $a,b \in \mathbb{Z}$ and $b>0$. Then there exist unique $q,r \in \mathbb{Z}$ so that $$a = bq+r$$ and $0 \leq r < b$.
We finished class by defining the notion of the greatest common divisor of two numbers integers.
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We started class by stating and proving a result called Bezout's identity. This tells us that $$\gcd(a,b) = \min\{ka + \ell b: k,\ell \in \mathbb{Z} \text{ and } ka+\ell b > 0\}.$$ One of the important consequences is that this gives us a way to argue when two elements are relatively prime by showing that $1$ can be written as an integral linear combination of these two elements. For instance, we used this to argue a result called Euclid's lemma: if $p$ is a prime number and $p \mid ab$, then it follows that either $p \mid a$ or $p \mid b$. We introduced the Euclidean algorithm as a method for computing the greatest common divisor of two numbers by repeatedly applying the division algorithm. One positive consequence of this algorithm is that it can be used to express the gcd of two numbers as an integral linear combination (as guaranteed by Bezout). We next discussed the notion of modular congruence, and said that addition and multiplication behave well under modular congruence.
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We started class today by reviewing the basics of induction. We stated that weak and strong induction are logically equivalent, and that each is logically equivalent to the well-ordering principle. We then used induction to prove that for all $n \geq 0$ we have $9 \mid 4^n+6n-1$.
We then defined the notion of an equivalence relation on a set $S$, and saw a few examples of relations that were equivalence relations. We defined equivalence class, and saw a theorem that tells us equivalence classes partition $S$.
We finished class by introducing some of the basic terminology about functions. For a function $\phi:A \to B$, we said that $A$ was the domain and $B$ was the codomain (also known as the target of $\phi$). The collection of all outputs of $f$ --- i.e., the set $\{\phi(a):a \in A\}$ --- is called the range (or image) of $f$. We said that if $\phi(A) = B$, then the function is surjective, and that if $\phi$ has the property that $\phi(a_1) = \phi(a_2)$ implies $a_1 = a_2$, then $\phi$ is called injective.
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We reviewed the definitions of injectivity, surjectivity and bijectivity today. We gave the proof templates for injectivity and surjectivity, defined the notion of function composition, and discussed various kinds of invertibility for functions. We showed that the composition of injective functions is injective, and stated a similar result for surjections.
Afterwards we defined the notion of a binary operation, and listed a number of examples (and non-examples) of some algebraically important sets and their binary operations. For example, we saw that matrix multiplication is a binary operation on $GL_n(\mathbb{R})$, but that matrix addition is not. A homework problem from Assignment 2 tells us that multiplication modulo $n$ is binary operation on $U(n)$, but a counterexample from class shows us that addition modulo $n$ is not. We then finished class by giving the definition of a group. A group is a set $G$ equipped with a binary operation $\star$ that satisfies three axioms:<
- (Associativity) If $g_1,g_2,g_3 \in G$ are given, then $(g_1 \star g_2) \star g_3 = g_1 \star (g_2 \star g_3)$
- (Identity) There exists $e_G \in G$ so that for all $g \in G$ we have $g \star e_G = e_G \star g = g$
- (Inverses) For every $g \in G$ there exists $h \in G$ so that $g \star h = h \star g = e_G$.
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Most of class today was spent discussing examples of groups. Many of these involved familiar sets ($\mathbb{Z}$, $\mathbb{R}^\times$, $GL_n(\mathbb{R})$ to name a few), but some of them were new to use (like $U(n)$, $\mathbb{Z}_n$ and $D_4$).
In the latter part of the class we discussed a few properties that all groups have. For instance, we gave a (very short) proof that there is only one element in a group which acts as the identity. We also stated a theorem that we called ``cancellation," and we used it to prove a variety of other results (e.g., that inverses are unique, a result that we informally described as the "walk like a duck" theorem, one that explains what happens when you take the inverse of an inverse, etc.). We finished class with the statement (and a sketch of a proof) of the socks and shoes theorem.
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In today's class we discussed a few properties that all groups have. For instance, we gave a (very short) proof that there is only one element in a group which acts as the identity. We also stated a theorem that we called ``cancellation," and we used it to prove a variety of other results (e.g., that inverses are unique, a result that we informally described as the "walk like a duck" theorem, one that explains what happens when you take the inverse of an inverse, etc.). We also discussed some standard notations in the discipline: the fact that some groups are written "multiplicatively" while others are written "additively," what it means to raise an element to a positive power. etc.) We finished class by discussing two interpretations for the word "order". When applied to a group, "order" simply means the number of elements in the group. When applied to an element of a group, it has a slightly more nuanced meaning. If there exists some $n>0$ so that $g^n = e$, then we say that $$|g| = \min\{k \in \mathbb{N}: g^k = e\}.$$ If there is no $n>0$ with $g^n=e$, then we say $|g| = \infty$. We computed the order of a few groups and a few elements. We saw that the (infinite) group $\mathbb{Z}$ has an element of infinite order, and we finished class by proving that any element of a finite group $G$ has order at most $|G|$.
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We defined the notion of subgroup. We spent the bulk of class stating and proving three theorems: the 1-step subgroup test, the 2-step subgroup test, and the finite subgroup test.
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We started class by using the $1$-step subgroup test to prove that $\text{SL}_n(\mathbb{R})$ is a subgroup of $\text{GL}_n(\mathbb{R})$. Afterwards we saw some examples of ``canonical" subgroups that are attached to any group. First, we defined the center of a group $G$, which we wrote as $Z(G)$; in short, the center of $G$ is the set of all elements of $G$ which commute with all other elements in $G$. More precisely, $$Z(G) = \{g \in G: hg = gh \text{ for all }h \in G\}.$$ For a given $g \in G$, we also defined the notion of the cyclic subgroup generated by $g$, as well as the centralizer of $g$.
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In today's class we answered two questions: if we have an element $a \in G$, what can we say about the order of a power of $a$ in terms of the order of $a$, and what can we say about the subgroup generated by a power of $a$ in terms of the subgroup generated by $a$? In service of these questions we determined what we can say about integers $i$ and $j$ when we know that $a^i = a^j$; this is the so-called Equal Power theorem. This was a key ingredient in telling us that $|a| = |\langle a \rangle|$, and indeed when $|a|=n$ that $\langle a \rangle = \{e,a,\cdots,a^{n-1}\}$. We then argued that $\langle a^k \rangle = \langle a^{\gcd(k,|a|)}\rangle$, and from this were able to conclude that $|a^k| = |a|/\gcd(k,|a|)$.
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We started class today by digging deep into the group $U(27)$. We showed that $\langle 2 \rangle = U(27)$, and then we used this to find other genreators of $U(27)$, as well as the orders for some other elements in the group. We then stated and (mostly) proved the fundamental theorem of cyclic groups. This theorem tells us everything there is to know about subgroups of a cyclic group. Combined with the equal subgroup theorem and the equal power theorem, we have essentially all the ingredients necessary to fully understand cyclic groups. We used this result to count the number of elements of a given order within a cyclic group. If $G = \langle a \rangle$ and $|a| = n$, then for any $d \mid n$ we know there are precisely $|U(d)|$ elements in $G$ with order $d$. We saw how this put restrictions on the size of $\{g \in G: |g| = d\}$ even when $G$ is not cyclic; for instance, since $|U(5)|=4$, we argued that there cannot be any group with precisely 10 elements of order $5$ (since $4 \nmid 10$).
We finished class by defining a permutation of a set, and ultimately argued that for a set $A$, the set of functions $$S_A = \{f:A \to A: f \text{ is a bijection}\}$$ is a group under the operation of function composition.
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In today's class we developed some of the basic notation that is used in investigating permutation groups. This notation is called cycle notation, and we saw how to encode a number of permutations as products of cycles. We determined how one computes products of cycles. We proved that $|S_n| = n!$. We showed that disjoint cycles commute. I handed out this handout for your personal enjoyment.
- Today's class was spent thinking more deeply about elements of symmetric groups. Most of our results fell into one of two camps: one was from the perspective of elements in disjoint cycle form, and the second was from the perspective of elements written as products of transpositions. For the former, we saw results which told us how to easily compute order via cycle structure, which we called Rufinni's theorem. Towards the latter we saw results concerning the ``evenness" or ``oddness" of elements from $S_n$, and we used these to define a subgroup called the alternating group (written $A_n$). The key result in this vein is that if we write the identity permutation as a product of transpositions, the number of transpositions which appear must be even; this allowed us to prove that for a given $\sigma \in S_n$, the parity of the number of transpositions in an expression of $\sigma$ always has a fixed parity (the result we called ``always even or always odd").
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We introduced the notion of isomorphism of groups. An isomorphism from $G$ to $\widehat{G}$ is a bijection $\phi: G \to \widehat{G}$ which satisfies $\phi(g_1 \star g_2) = \phi(g_1)\bullet \phi(g_2)$ for all $g_1,g_2 \in G$, where here $\star$ denotes the operation in the (domain) group $G$ and $\bullet$ denotes the operation in the (codomain) group $\widehat{G}$; if an isomorphism from $G$ to $\widehat{G}$ exists, we say that $G$ is isomorphic to $\widehat{G}$, and we write $G \simeq \widehat{G}$ (or $G \approx \widehat{G}$). We saw some explicit examples of groups that are isomorphic, as well as some examples of groups that are not isomorphic (e.g., $\mathbb{R} \not\approx \mathbb{R}^\times$). We stated that the notion of ``isomorphic to" is an equivalence relation on the set of groups.
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Most of our class today was spent discussing properties that isomorphisms have. For instance: how are qualities of a given element effected when it is passed through an isomorphism. In class today we saw that --- for example --- the order of an element and the order of its image under an isomorphism must be the same; that is, if $\phi:G \to \hat G$ is an isomorphism and $a \in G$, then $|a| = |\phi(a)|$. We also saw that $\phi(e_G) = e_{\hat G}$, as well as a few other properties that tell us how isomorphisms behave on elements. We also analyzed how isomorphisms act on groups as a whole. For example, if $G$ and $\hat G$ are isomorphic, then $G$ is abelian if and only if $\hat G$ is abelian, and $G$ is cyclic if and only if $\hat G$ is cyclic (even more, if $\langle g \rangle = G$, then $\langle \phi(g) \rangle = G$). One of the very useful properties about isomorphism is that it preserves the number of elements of a given order. This means that if we have two groups $G$ and $\hat G$ and some $n \in \mathbb{N}$ so that the number of elements of order $n$ in $G$ differs from the number of elements of order $n$ in $\hat G$, then we know that $G \not\simeq \hat G$. We finished class by stating a theorem that tells us that any cyclic group of order $n<\infty$ is isomorphic to $\mathbb{Z}_n$, and any cyclic group of infinite order must be isomorphic to $\mathbb{Z}$.
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We started class today by proving the result about cyclic groups that was stated at the end of class last time. We then went on to define a special kind of isomorphism on a group $G$, namely an inner automorphism. If $g \in G$ is a given element, then the function $\phi_g: G \to G$ given by $\phi_g(x) = gxg^{-1}$ is called the inner automorphism defined by $g$; we proved that this is an isomorphism. We then showed that $\phi_{g_1g_2} = \phi_{g_1} \circ \phi_{g_2}$, which looked an awful lot like a statement that the function $\psi:G \to \text{Inn}(G)$ was a homomorphism of groups. To move towards that, we thought a bit more broadly: we proved that the set $$\text{Aut}(G) = \{f: G \to G | f \text{ is an isomorphism}\}$$ is a group under composition. We then finished class by stating the isomorphism $\text{Aut}(\mathbb{Z}_n) \simeq U(n)$, and we got about halfway through its proof.
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We started class today by finishing our proof of $\text{Aut}(\mathbb{Z}_n) \simeq U(n)$. Afterwards we stated and sketched a proof of Cayley's theorem: any (finite) group $G$ can be viewed as a subgroup of some symmetric group. We preceded this with an example of how one can use elements of $S_4$ to represent $D_4$. In many ways, Cayley's theorem is a stark reminder that one cannot be willing to discard the notion of ``groups of functions" when studying group theory, since all groups can be thought of as sitting inside some group of functions.
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Today's class was mostly spent picking up some facts about homomorphisms that follow quickly from ideas we thought of when introducing the notion of isomorphism. We examined how homomorphisms act on both elements and subgroups. When $\phi: G \to \hat G$ is a homomorphism, we introduced the notion of the image of a subgroup $H \leq G$ (which we wrote $\phi(H)$ and proved is a subgroup of $\hat G$) as well as the inverse image of a subgroup $K \leq \hat G$ (which we wrote $\phi^{-1}(K)$ and proved is a subgroup of $G$). In particular, we defined the kernel of $\phi$ --- denoted $\ker(\phi)$ --- to be $\ker(\phi) = \phi^{-1}(\{e_{\hat G}\})$, and the image of $\phi$ to be $\phi(G)$. We proved that inverses images of elements in $\phi(G)$ are "shifts" of the kernel (in the sense that if $\phi(g) = \hat g$, then $\phi^{-1}(\{\hat g\}) = g\ker(\phi) = \{gk: k \in \ker(\phi)\}$). This was then used to show that when $\ker(\phi) = \ell$, then the map $\phi: G \to \hat G$ is $\ell$-to-$1$. As a consequence, we have a group-theoretic version of the rank nullity theorem from linear algebra: if $\phi:G \to \hat G$ is a homomorphism, then $|\phi(G)||\ker(\phi)| = |G|$.
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Today we started with a "warm up" exercise to put some of the ideas from last class into practice. We did this by computing the kernel of the natural homomorphism from $G$ to $\text{Inn}(G)$. We spent the rest of class building some theory which will allow us create old groups out of new groups. That new idea was the notion of direct sum: if $(G_1,\star)$ and $(G_2,\bullet)$ are groups, then we defined $$G_1 \oplus G_2$$ to be the group whose underlying set is $$\{(g_1,g_2): g_1 \in G_1,g_2 \in G_2\}$$ and whose operation is given by $$(g_1,g_2)(\hat g_1,\hat g_2) = (g_1 \star \hat g_1,g_2 \bullet \hat g_2).$$ It is a theorem that when defined in this way, $G_1 \oplus G_2$ is a group. We discussed the ways in which direct sum is almost a group operation on the set of groups; strictly speaking it fails almost all axioms if we think of two groups as "the same" only when they are precisely equal, but satisfies all axioms except inverse if we instead loosen our interpretation of when groups are "the same" to mean when they are isomorphic. For instance, we showed that $\{e\} \oplus G \simeq G$, and so the trivial group acts like an identity element under this operation (when we think of groups "up to isomorphism"). We saw that $|G_1 \oplus G_2| = |G_1||G_2|$, the identity in $G_1 \oplus G_2$ is the element $(e_{G_1},e_{G_2})$, the inverse of an element $(g_1,g_2)$ is given by $(g_1^{-1},g_2^{-1})$. We also saw how to compute the order of an element in a direct sum, and used this to count the number of elements of order $10$ in $D_4 \oplus \mathbb{Z}_{10}$. We finished class by giving results that tell us when direct sums are either cyclic or abelian.
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One of the big questions in abstract algebra is to determine the degree to which one can express a "complicated" group in terms of "combinations" of "simpler groups." All of this is a bit fuzzy: what should count as a "simpler group," and what does it mean to "combine" two groups into another group? We'll see several ways of answering these questions as we move through the semester, but the direct sum operation is at least one good candidate for what it might mean to "combine" two groups. So we ask: when is it possible to write a group as a direct sum of groups of the form $\mathbb{Z}_n$? We answered this question in the context of certain kinds of groups today. We started with an example where we computed (with some effort!) that $\text{Inn}(G) \simeq \mathbb{Z}_2 \oplus\mathbb{Z}_2$. Building on the theorem from last class which tells us when the sum of cyclic groups is cyclic, we were able to use this to argue that for $n = m_1 \cdots m_k$ we get $$\mathbb{Z}_n \simeq \mathbb{Z}_{m_1} \oplus \cdots \oplus \mathbb{Z}_{m_k} \quad \text{ if and only if }\quad \gcd(m_i,m_j) = 1 \text{ for all }i \neq j.$$ We also saw that a similar statement holds for $U$-groups, namely if $n = m_1\cdots m_k$ where $(m_i,m_j) = 1$ for all $i \neq j$, then $$U(n) \simeq U(m_1) \oplus \cdots \oplus U(m_k).$$ This, combined with a result of Gauss, allowed us a technique to express any $U$-group as a direct sum of groups of the form $\mathbb{Z}_\ell$. We used this, for instance, to the group $\text{Aut}(\mathbb{Z}_{100})$, and even to show that $U(n)$ is cyclic if and only if $n \in \{1,2,4\}$ or $n$ is a power of an odd prime or $n$ is twice a power of an odd prime.
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We started class today by using Gauss' theorem on $U(p^k)$ to prove that when $n = p_1^{e_1} \cdots p_k^{e_k}$, one has $$|U(n)| = \prod_{i=1}^k p_i^{e_i-1}(p_i-1).$$
Afterwards, we introduced the notion of (left and right) cosets of a subgroup $H$ inside a group $G$. We saw some examples of cosets (for $\{1,-1\} \leq Q_8$, for $\langle (1,2,3) \rangle \leq S_4$, for $\langle (1,2) \rangle \leq S_3$, and finally for $\langle 3 \rangle \leq \mathbb{Z}_{12}$), and we noticed some properties from these examples. We gave a result (which we called "Properties of cosets") which detailed several critical facts about cosets, and then we stated Lagrange's Theorem. This results says that if $H$ is a subgroup of $G$, then $|H| \mid |G|$, and that in fact the number of (left) cosets of $H$ in $G$ is $|G|/|H|$.
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Today we proved Lagrange's theorem and we saw some of its immediate applications. Some of the most frequently used are the fact that the order of any element $a \in G$ must divide $|G|$, and that any group of prime order must be isomorphic to $\mathbb{Z}_p$. For instance, we saw some famous results in number theory (Euler's Theorem and Fermat's little theorem) which are immediate consequences of the fact that $|a| \mid |G|$. We proved that there are only two groups of order $4$ (up to isomorphism).
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We saw two more applications of Lagrange's theorem today. The first was that one can tweak Fermat's little theorem slightly so that it applies for any "base" (not just those which are relatively prime to $p$), and that one can use this as a way to check when certain numbers are not prime. For instance, we proved that $4$ is not prime using this idea without ever giving a factorization of $4$. We then used Lagrange's theorem to characterize those groups of order $2p$, where $p$ is an odd prime. We saw the two possibilities for a group of this size are $\mathbb{Z}_{2p}$ and the so-called dihedral group of order $2p$, which we denoted $D_p$.
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Today we considered normal subgroups, which are subgroups $H$ of a group $G$ such that $aH = Ha$ for all $a \in G$. We saw a number of examples of normal subgroups; for instance, any subgroup of an abelian group is normal, and $\text{SL}_n(\mathbb{R})$ is normal in $\text{GL}_n(\mathbb{R})$. We saw a theorem which helps determine when a subgroup is normal (the so-called normal subgroup test).
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We started class today by discussing normal subgroups a bit more. We proved that any index two subgroup (i.e., any subgroup $H \leq G$ for which there are precisely two cosets) is normal. We then used this to argue that $A_n \lhd S_n$, and also as an instrumental tool in proving that all subgroups of $Q_8$ are normal. Along the way we stated the fact (but did not prove) that for any group we have $\{e\} \lhd G$ and $G \lhd G$.
After this we asked whether there was a reasonable way to put a binary operation on the set $G/H$ of left costs of $H$ in $G$. We proved that the natural operation $$(aH)(bH) = (ab)H$$ is well-defined if and only if $H \lhd G$. We then proved that when $H \lhd G$, the set $G/H$ equipped with this operation is a group. The identity element is the trivial coset $eH = H$, and for a given coset $aH \in G/H$ we have $(aH)^{-1} = a^{-1}H$.
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Today we discussed the group $G/H$ in depth. We commented that $G/H$ is useful to study because it is intimately connected to the group $G$, but is smaller (and therefore easier to study). We thought about the group $G/Z(G)$ and saw on the one hand that this group is isomorphic to $\textrm{Inn}(G)$, and on the other hand that if $G/Z(G)$ is cyclic then $G$ is abelian. We also considered how normality interacts with homomorphisms, namely that pull-backs (under a homomorphism) of normal subgroups from the codomain are normal in the domain, whereas push-forwards of normal subgroups from the domain are normal in the image (not necessarily normal in the codomain!). This tells us that, for instance, $\ker(\phi) = \phi^{-1}(\{e\})$ is normal in $G$, since $\{e\} \lhd Q$. We said even more: a subgroup $N$ is normal in $G$ if and only if $N$ is the kernel of some homomorphism.
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Last class we proved that if $f:G \to \hat G$ is a homomorphism, then $\ker(f) \lhd G$. Today we asked: what does the group $G/\ker(f)$ represent? We were able to prove the so-called Fundamental theorem of homomorphisms: $$\frac{G}{\ker(f)} \simeq \text{im}(f).$$ We then applied this theorem to a few special cases. For instance, when $f:G \to \text{Inn}(G)$ is the natural map, we saw that the Fundamental theorem of homomorphisms gives $G/Z(G) \simeq \text{Inn}(G)$; this is a result we saw a few lectures ago, but the fundamental theorem gives us a new perspective on this identity. We also used the first isomorphism theorem to show that $\text{GL}_n(\mathbb{R})/\text{SL}_n(\mathbb{R}) \simeq \mathbb{R}^\times$, and also that $\mathbb{Z}/\langle n \rangle \simeq \mathbb{Z}_n$.
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Today we introduced the primary decomposition form of the fundamental theorem of finite abelian groups. We computed several examples to see how one goes about enumerating the possible group structures for an abelian group of size $n$. We stated that this theorem is primarily useful in that it provides a list of possible group structures that a group might take on; one can then use "process of elimination" to argue that a given group is isomorphic to something, instead of having to produce an explicit isomorphism. We used the theorem to show that if $n$ is a squarefree natural number, then the only abelian group of order $n$ is $\mathbb{Z}_n$.
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We started today continuing our discussion of the fundamental theorem of finite abelian groups. This began with a statement of the invariant factor form of the fundamental theorem of finite abelian groups. We saw how to put this theorem into practice for groups of a few sizes, and we also saw how the primary decomposition and invariant factor forms are related to each other.
We did give a rough outline for how one proceeds to prove the fundamental theorem of finite abelian groups, and even provided a few details. For instance, we proved that if $N_1,N_2 \lhd G$ so that $N_1N_2 = G$ and $N_1 \cap N_2 = \{e_G\}$, then $G \simeq N_1 \oplus N_2$. [Note: this is true even when $G$ is not abelian.]
At the end of class we spent a little time talking about where group theory "goes next," and even saw a paper which provided a classification of groups of order $p^6$.
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Today we introduced the notion of rings, beginning with a discussion of their axioms, some basic examples, and terminology associated to rings. For instance, we defined units and zero-divisors; we saw that many familiar mathematical sets are rings (like $\mathbb{Z}$, $M_n(\mathbb{R})$, and $\mathbb{Z}[x]$).
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We started today with a few more examples of rings, including some that are perhaps not so familiar (like Hamilton's quaternions). We proved some of the properties that are common to all rings; for instance, for any $a \in R$ we have $a 0_R = 0_R a = 0_R$. We also proved a number of results about how additive inverses behave under multiplication. We then went on to introduce the notion of a subring; we saw that there is a subring test which are analogous to the subgroup tests we've already studied (but which are slightly more involved to allow for the richer structure that a ring possesses). This test lets us determine whether a given subset of a ring is a subring without checking all axioms of a rin. We discussed a number of canonical subrings. We introduced the notion of integral domain.
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A ring is called an integral domain if it is commutative, has a multiplicative identity, and has no zero divisors. Integral domains were the topic of discussion today. We gave a few examples of rings that are integral domains, as well as rings that are not integral domains. We saw that integral domains belong to a class of rings which satisfy the cancellative property: if $a \neq 0_R$ and $ab = ac$, then in fact $b = c$. Note that this is much like the cancellative property in groups, but the proof we offered in the integral domain setting --- which relies on us subtracting a quantity from both side and then un-distributing a multiplication --- is quite far from the proof we offered for groups. We remarked that when $a \in U(R)$, the proof that $ab = ac$ is essentially identical to the proof we gave for groups, so part of the magic of this result is that we arrive at the same conclusion with weaker information. We defined fields to be integral domains for which $U(R) = R \setminus \{0_R\}$, and we saw that all finite integral domains are fields; the proof of this result was very much in the spirit of the finite subgroup test, even though the stated result seems to be quite different from this familiar theorem.
In the latter part of the class we introduced the notion of characteristic of a ring. If there exists a positive integer $n$ so that the $n$-fold sum of any element $r \in R$ is $0_R$, then we define the characteristic of $R$ to be the minimal value among all such $n$. If there is no such $n$, then we say that $R$ has characteristic $0$. We commented that the characteristic of the ring is simply the exponent of the group $(R,+)$. We proved that if $1_R \in R$, then $\text{char}(R) = |1_R|$. We showed that if $R$ is an integral domain, then $\text{char}(R)$ is either $0$ or prime. We commented that for any prime $p$ and any $n \in \mathbb{N}$, there exists a unique field $\mathbb{F}$ with $|\mathbb{F}|=p^n$.
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In today's class we defined the notion of an ideal: this is simply a subring which has some additional properties. We commented (but didn't make precise) that ideals are the ring-theoretic analogs of normal subgroups. We spent most of the class period discussing examples of ideals. Among the many examples of such ideals, we discussed the notions of principle and multiply-generated ideals. We finished class by showing that any ideal of $\mathbb{Z}$ is principle; indeed, if $I$ is a nontrivial ideal of $\mathbb{Z}$ and $a = \min I \cap \mathbb{N}$, we showed that $I = \langle a \rangle$. This gives us a new way to think about Bezout's identity from an ideal theoretic perspective; in this context, Bezout says that if $a$ and $b$ are integers which are not both zero, then $\langle a,b\rangle = \langle \gcd(a,b)\rangle$.
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We started class today by considering a particular ideal inside of $\mathbb{Z}[x]$; the ideal was $\langle x,2\rangle$, which we showed was equivalent to the set of polynomials in $\mathbb{Z}[x]$ whose constant term is even. We proved that this ideal is not principle, and hence that $\mathbb{Z}[x]$ doesn't have the "PID" property that $\mathbb{Z}$ enjoys.
Building on the idea that ideals are the ring-theoretic analogs of normal subgroups, we then defined the notion of a quotient ring $R/I$ in the case that $I$ is an idea of $R$. The elements of $R/I$ are the (additive) cosets of $I$ with representatives in $R$ (i.e., a generic element of $R/I$ takes the form $r+I$ for some $r \in R$), and the addition and multiplication operations are defined by \begin{align*} (r+I)+(s+I) &= (r+s) + I\\ (r+I)(s+I) &= rs+I. \end{align*} We stated that these operations are well-defined if and only if $I$ is an ideal. The point of studying quotients for rings echoes what we've already seen in the group theoretic context: quotients of a given ring are often "simpler" than the original ring (in that there are "fewer" elements), and the structure of these quotients often encodes something meaningful about the ring itself.
For example, we defined the notion of "prime" for an ideal, and we argued that for a commutative ring $R$ with $1_R \in R$, an ideal $I$ is prime if and only if $R/I$ is an integral domain. We used this to argue that $\langle x, 2 \rangle \subseteq \mathbb{Z}[x]$ is a prime ideal, since we proved that it's quotient was precisely $\{0+\langle x,2 \rangle,1+\langle x,2\rangle\}$, and this set is an integral domain. (Indeed, it's "just like" $\mathbb{Z}_2$, which is a field.)
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We started class today by reviewing our analysis of the quotient ring $\mathbb{Z}[x]/\langle x,2\rangle$, including why it's given by precisely $\{0+\langle x,2\rangle,1+\langle x,2\rangle\}$, and why it's an integral domain. We built on these ideas to think about a related quotient: $\mathbb{Z}[x]/\langle x\rangle$. We saw that $\{z+\langle x \rangle: z \in \mathbb{Z}\}$ give a complete, non-redundant set of cosets for the quotient (in that every coset of $\mathbb{Z}[x]/\langle x\rangle$ is equal to some coset of that form, and no two cosets of that form are equal), so that this collection was "just like" $\mathbb{Z}$. Since $\mathbb{Z}$ is an integral domain, this means that $\langle x \rangle$ and $\langle x,2 \rangle$ are both prime. But since $\mathbb{Z}[x]/\langle x,2\rangle$ was actually a field (and not just an integral domain), we wondered: does the ideal $\langle x,2 \rangle$ have some special property that makes its quotient so much nicer than "just" an integral domain? Said another way: is there something more than just "primeness" in this ideal?
We saw this question has a positive answer through the definition of maximal ideal. Maximal ideals $I \subseteq R$ are those proper ideals of $R$ which have no proper ideal "between" themselves and $R$: if $I$ is maximal and $I \subsetneq J \subseteq R$, then we must have $J = R$. We proved that $\langle x,2\rangle$ is maximal directly, but we also proved that in a commutative ring with unity, $I$ is a maximal ideal if and only if $R/I$ is a field. Since we saw that $\mathbb{Z}[x]/\langle x,2 \rangle$ is "just like" the field $\mathbb{Z}_2$, then, we have another proof that $\langle x,2\rangle$ is maximal. We used this result to show that if $R$ is a commutative ring with unity, then any maximal ideal is prime.
To conclude class we finally made good on the definition that we've been waving our hands over for a few lectures concerning isomorphism. We defined the notion of ring homomorphism and isomorphism.
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The first half of class today was spent going through some examples of ring homomorphisms, as well as some basic properties of ring homomorphisms. Most of the properties we saw were analogous to results we have in the group-theoretic context; the one exception is that if $1_R \in R$ exists and $f:R \to S$ is a ring homomorphism, then $f(1_R) = 1_{f(R)}$; this is less strong then the group theoretic result, essentially because we don't have walks like a duck for multiplicative identities in rings. We stated that a subring is an ideal if and only if its the kernel of a homomorphism, and we stated the fundamental theorem of ring homomorphisms.
We went on to being an analysis of the ring $\mathbb{R}[x]/\langle x^2+1\rangle$. In class we were able to prove that $\{ax+b+\langle x^2+1\rangle: a,b \in \mathbb{R}\}$ is a complete, non-redundant set of cosets, and we also observed that $(x+\langle x^2+1\rangle)^2 = -1+\langle x^2+1\rangle.$ With this as motivation, we suspected that perhaps $\mathbb{R}[x]/\langle x^2+1\rangle \simeq \mathbb{C}$. We'll prove this result next class.
I also made some cookies, and some folks asked for recipes. Both come from Smitten Kitchen cookbooks. The pretzel cookie was the cookie base of the "pretzel Linzers with Salted Caramel" recipe from Smitten Kitchen Every Day (though the recipe isn't on the Smitten Kitchen blog, it does seem to be published in a few places like this). The chocolate cookie was the "brownie roll-out cookie" from the Smitten Kitchen Cookbook and is available on the Smitten Kitchen Blog here.
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We finished our construction of the complex numbers as the quotient $\mathbb{R}[x]/\langle x^2+1\rangle$ to start off the class, and in particular showed the isomorphism by using the fundamental theorem of homomorphisms (for rings). This led to a larger question: to what degree can the ideas from this example be extended to other fields? We know (from a homework) that if $F$ is a field, then $F[x]$ is a Euclidean domain, and hence we get $F[x]/\langle p(x) \rangle$ is a field if and only if $\langle p(x) \rangle$ is maximal. So our question was: what about a polynomial $p(x)$ makes the ideal it generates maximal? We defined the notion of irreducible polynomial, and ultimately proved that a polynomial $p(x) \in F[x]$ is irreducible if and only if $\langle p(x)\rangle$ is maximal.
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To wrap up the course we asked a fairly broad question: what's so algebraic about what we've done during the course of the semester. We saw that there were three big avenues of mathematical research which were all conversant with the objects we've discussed during the semester, and that abstract algebra was born our of attempts to distill these ideas into some coherent structure that could describe the phenomena that were being observed in these various contexts. We also spent some time talking a bit about what (typically) comes next in abstract algebra: Galois theory. I said one sentence about my Galois theoretic research.