1. the definition of an operation, $f\colon A^n\to A$,
2. the definition of the arity of an operation, $\alpha(f)\in \mathbb{N}$,
3. the fact that there are operations of arity $0$,
4. the definition of a signature, $\sigma=(F,\alpha)$,
5. the definition of an algebra of signature $\sigma$, $\mathbb{A}=\langle A; F^{\mathbb{A}}\rangle$,
6. the definition of the universe, $A$, of an algebra $\mathbb{A}=\langle A; F^{\mathbb{A}}\rangle$.

• $\langle \mathbb{N}; S(x), 0\rangle$,
• $\langle \mathbb{N}; +, \cdot, S(x), 0\rangle$,
• $\mathbb{B}_2=\langle \{0,1\}; \wedge,\vee,\neg,0,1\rangle$.

1. Let $A$ be an army and for $x,y\in A$ let $\diamondsuit(x,y)$ = the least common commander of $x$ and $y$. $\diamondsuit$ is an operation on $A$, hence $\langle A; \diamondsuit\rangle$ is an algebra. (This kind of algebra can be used to model any hierarchical structure where any two elements have a least common upper bound.)
2. Any scheme for merging ranked lists of voter preferences is an operation on the set of ranked lists.
3. Procedures for combining data structures (insertion, deletion, merging) are operations on the class of data structures.

We let $S$ be a set and $\mathcal{F}=\{f\;|\;f\colon S\to S\}$ be the set of functions from $S$ to $S$. Let $1=\textrm{id}_S$ be the identity function from $S$ to $S$. We argued that the function algebra, $\langle \mathcal{F}; \circ, 1\rangle$, is a structure with an associative multiplication $\circ$ and a unit element $1$. (The associative law is $(\forall f)(\forall g)(\forall h)(f\circ (g\circ h)=(f\circ g)\circ h)$) and the unit laws are $(\forall f)(1\circ f=f)$ and $(\forall f)(f\circ 1=f)$). We defined the class of monoids to be the class of all algebraic structures $\langle M; \circ, 1\rangle$ that have an associative multiplication and a unit element. Monoids will turn out to be the correct algebraic model for functions under composition. We will be able to use monoids to prove that the universally quantified laws of functional composition which involve the identity function are exactly the consequences of the associative law and the unit laws.

We reviewed this handout on functions, and then introduced some function-related terminology for algebra, namely:

1. homomorphism,
2. embedding,
3. isomorphism
4. subuniverse/subalgebra

We discussed these slides, which prove the Cayley Representation Theorem for monoids.
Quiz 1.

1. A semigroup is an algebra $\langle S; \circ\rangle$ with a single associative binary operation. Semigroups is the “correct” algebraization of functions under composition, $\langle \textrm{Funct}(X,X); \circ\rangle$.
2. A left cancellative semigroup is a semigroup $\langle S; \circ\rangle$ with a single associative binary operation which also satisfies \[(\forall f)(\forall g)(\forall h)((f\circ g=f\circ h)\to (g=h)). \] Left cancellative semigroups are the “correct” algebraization of injective functions under composition.
3. A right cancellative semigroup is a semigroup $\langle S; \circ\rangle$ with a single associative binary operation which also satisfies \[(\forall f)(\forall g)(\forall h)((f\circ h=g\circ h)\to (f=g)). \] Right cancellative semigroups are the “correct” algebraization of surjective functions under composition.
4. A group is an algebraic structure $\langle G; \circ, {}^{-1},1\rangle$ with a binary operation $x\circ y$ for which the associative law holds, a unary operation $x^{-1}$ for which the inverse laws hold, and a constant $1$ for which the identity laws hold. Groups are the “correct” algebraization of bijective functions under composition.

We discussed examples of groups.

Quiz 2.

We defined some words (function, map, bijection, invertible map, permutation). We noted that the set of permutations of a set is closed under composition, inversion, and contains the identity function, and this fact suggested the definition of a group. We discussed different signatures for groups and different operation symbols for groups (e.g., multiplicative notation versus additive notation).

The group of all permutations of a set $X$ is called the symmetric group on $X$. We examined the Cayley Representation for monoids and found that it extends to a representation for groups. This shows that every group is embeddable in a symmetric group.

We tried to determine all small groups up to isomorphism, but we did not get far. We only showed that there is 1 group of size 1 (up to isomorphism) and 1 group of size 2 (up to isomorphism).

We classified groups up to size $4$. We then began discussing symmetric groups, $\langle S_X; \circ, {}^{-1},1\rangle$, and practiced multiplying permutations following this handout.

We started with a practice problem whose goal was to show that $|S_n|=n!$. We then explained why, since $|S_2|=2!=2$, the Cayley Representation Theorem proves that if $\mathbb{G}$ is a $2$-element group, then $\mathbb{G}\cong S_2$. This provides a new proof that there is only one group of size $2$ up to isomorphism. (The first proof we saw was by “brute force”.)

Next, we defined subalgebra (notation $\mathbb{B}\leq \mathbb{A}$) and spent the rest of the class discussing examples of subgroups (notation $\mathbb{H}\leq \mathbb{G}$).

1. $\langle \mathbb{Z}; +,-,0\rangle$ is a subgroup of $\langle \mathbb{R}; +,-,0\rangle$.
2. $\textrm{GL}_n(\mathbb{R})$ (general linear group) has subgroups
   • $\textrm{SL}_n(\mathbb{R})$ (special linear group)
   • $\textrm{O}_n(\mathbb{R})$ (orthogonal group)
   • $\textrm{SO}_n(\mathbb{R})$ (special orthogonal group).
3. Every group is isomorphic to a subgroup of the symmetric group.

1. Cyclic groups $C_n=\langle \{1, r, r^2, \ldots, r^{n-1}\}; \circ, {}^{-1}, 1\rangle$. ($|C_n|=n$)
2. Dihedral groups $D_n=\langle \{1, r, r^2, \ldots, r^{n-1}, f, rf, \ldots, r^{n-1}f\}; \circ, {}^{-1}, 1\rangle$. ($|D_n|=2n$)

Quiz 3.

• We reviewed the groups $C_n = \langle r\;|\;r^n=1\rangle$ and $D_n = \langle r, f\;|\;r^n=1, f^2 = 1, fr=r^{-1}f\rangle$.
• We defined the product, $H\times K$, of groups $H$ and $K$.
• We stated, without proof, the classification of groups of order at most 10 up to isomorphism. All such groups are either (i) products of cyclic groups, (ii) dihedral groups, or (iii) the 8-element quaternion group, $Q_8$.
• We discussed the fact that the collection of subgroups of $G$, ordered by inclusion, is a lattice. The greatest lower bound $H\wedge K$ of two subgroups $H, K\leq G$ is their intersection $H\cap K$. The least upper bound $H\vee K$ of two subgroups $H, K\leq G$ is the subgroup generated by their union $\langle H\cup K\rangle$.
• We used UACalc to examine some subgroup lattices (e.g. $C_2^3$ and $D_8$).
• We reviewed terminology for functions with the aim of lifting the terminology to groups and other algebraic structures.

We started by reviewing terminology for functions. After this, we lifted the terminology to groups. The discussion was illustrated with the example of the homomorphism $h\colon D_3\to C_4$ defined by $h(\textrm{rotation}) = 1$ and $h(\textrm{nonrotation}) = r^2$. For this choice, $\textrm{im}(h)=\{1,r^2\}\subseteq C_4$, $\textrm{ker}(h)=\{1,r,r^2\}\subseteq D_3$, and $\textrm{coim}(h)=\{\{1,r,r^2\}, \{f, rf, r^2f\}\}=\{[1],[f]\}$.

We explained some special properties of algebra homomorphisms (in particular, group homomorphisms) following these slides.

Test your vocabulary!

We started with a student question about “lattices”. We explained why the class of partially ordered sets, $\langle P; \leq \rangle$, equipped with all order-preserving maps is not “algebraizable”, since there exists bijective morphisms of posets that are not isomorphisms. But the related class of lattices, $\langle L; \vee, \wedge\rangle$, equipped with lattice homomorphisms is algebraizable and captures much of the category of partially ordered sets.

Next, we returned to our main thread, which is the study of homomorphisms. We examined these familiar examples:

1. (absolute value) $$A \colon \langle \mathbb{R}^{\times};\cdot, {}^{-1},1\rangle\to \langle \mathbb{R}^{\times};\cdot, {}^{-1},1\rangle\colon x\mapsto |x|. $$
2. (exponential function #1) $$E_1 \colon \langle \mathbb{R};+,-,0\rangle\to \langle \mathbb{R}_{>0};\cdot, {}^{-1},1\rangle\colon x\mapsto e^x. $$ $E_1$ is an isomorphism! The inverse of $E_1(x)$ is $L_1(x) := \textrm{ln}(x)$.
3. (exponential function #2) $$E_2 \colon \langle \mathbb{R};+,-,0\rangle\to \langle \mathbb{C}^{\times};\cdot, {}^{-1},1\rangle\colon \theta\mapsto e^{i\theta}. $$ We explained why $E_2$ is a homomorphism using Euler's Formula and the Angle Addition Formulas.

Quiz 4.

We discussed the Canonical Factorization of a Homomorphism following these slides

We completed the slides on the Canonical Factorization of a Homomorphism. We made the following additional observations:

1. If $h\colon \mathbb{A}\to \mathbb{B}$ is a homomorphism and $\mathbb{S}$ is a subalgebra of $\mathbb{A}$, then $h(\mathbb{S})$ is a subalgebra of $\mathbb{B}$. (The image of a subalgebra is a subalgebra.)
2. If $h\colon \mathbb{A}\to \mathbb{B}$ is a homomorphism and $\mathbb{T}$ is a subalgebra of $\mathbb{B}$, then $h^{-1}(\mathbb{T})$ is a subalgebra of $\mathbb{A}$. (The preimage of a subalgebra is a subalgebra.)

Next we determined all components of the Canonical Factorization of a Homomorphism in these cases:

1. $h\colon D_3\to C_4$ where $h(\textrm{rotations})=1$ and $h(\textrm{flips})=r^2$.
2. $\textrm{AbsVal}\colon \mathbb{R}^{\times} \to \mathbb{R}^{\times}\colon x\mapsto |x|$.

We discussed why groups are interesting, both within mathematics and within algebra. We defined cosets of subgroups. We compared coimages, kernels, and Kernels. We worked on this handout.

Quiz 5.

We continued our discussion of cosets of subgroups following these slides.

We discussed Lagrange's Theorem following these slides. I circulated this midterm review sheet.

Quiz 6.

• the Cauchy number is easy to calculate and tells us whether a permutation is even or odd.
• The set of even permutations in $S_n$ is the Kernel of $\textrm{sgn}$. It is a normal subgroup of index $2$ in $S_n$. This subgroup is called the Alternating Group, $A_n$.
• The sign homomorphism is needed in Linear Algebra to compute the determinant using the Leibniz formula: $\det([a_{i,j}])=\sum _{\tau \in S_{n}}\textrm{sgn}(\tau )\prod _{i=1}^{n}a_{i,\tau(i)}$.

We next discussed the operation $\gamma_g\colon x\mapsto g^{-1}xg =: x^g$ of conjugacy by $g$. We verified that conjugacy by $g\in G$ is an automorphism of $G$. An automorphism of $G$ is called inner if it agrees with conjugacy by $g$ for some $g\in G$.

We stated our next goals, which are to show that: $A_n$ (= the Alternating Group on $n$ points = the subgroup of even permutations in $S_n$) is normal in $S_n$ and satisfies $[S_n:A_n]=2$; $A_n$ is simple if $n\geq 5$. We began a discussion of why simple groups are important in the study of finite groups. This discussion included stating the definition of a short exact sequence of groups: $$ 1\to K\stackrel{i}{\to} G\stackrel{\mu}{\to} G/K\to 1. $$ We began a discussion about how such a sequence leads to a product-like decomposition of $G$. We pointed out that simple groups are exactly those nontrivial groups that cannot be properly decomposed this way.

Quiz 7.

• The smallest nonabelian simple group is $A_5$ and $|A_5|=60$.
• All finite simple groups of order $<6000$ are isomorphic to either (i) $\mathbb Z_p$ for some prime, (ii) $A_n$ for some $n\geq 5$, or (iii) $\textrm{PSL}_n(\mathbb{F})$ for some $n$ and some field $\mathbb{F}$.
• Galois proved the simplicity of $A_5$ in 1831, Jordan proved the simplicity of $A_n$, $n\geq 5$ in 1870.
• Serious attempts to classify the finite simple groups began in 1963 with the proof of the Feit-Thompson Theorem. The Classification of Finite Simple Groups was claimed to be completed in 1983, but a gap was found and we now date the CFSG to 2004. The CFSG has not been fully verified by computer, but the Feit-Thompson Theorem part verified by computer in 2012.