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\fancyhead[CO,CE]{Math 120: Definitions from \emph{Dummit and Foote}}
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\begin{document} 

\title{Math 120: Definitions from \emph{Dummit and Foote}}
\author{Lucas Garron}
\maketitle

Key terms and definitions.\\
Many things are left implicit, e.g. $N \tleq G$ also implies $G$ is a group.

\def\n{\newline}
\def\e{\\[3mm]\hline\\[3mm]}

\begin{mylist}{new Document}

\item[Functions (2)] injective, surjective, bijective

\item[Relations (3)] reflexive, symmetric, transitive, equivalence (all three), equivalence classes, partitions

\item[$\Z / n \Z$ (9)] Integers modulo $n$ (with addition and multiplication)

\item[$(\Z / n \Z)^\times$ ( 10)] Relatively prime residue classes

\item[Operations (16)] Binary, associative, commutative

\item[Group (16-17)] An ordered pair $(G, \star)$ with:

\begin{enumerate}
\item $G$ closed under $\star$
\item $\star$ associative (on $G$)
\item Identity: $e \in G$
\item Inverse: $a \in G \imp \E a\i \in G$
\end{enumerate}

\item[Abelian (17)] $(G, \star)$ is abelian/commutative if $\star$ is commutative

\item[Cancellation (20)] e.g. left cancellation: $au = av \imp u=v$

\item[Order (20)] The order of $x \in G$ is the smallest $n \in \Z^+$ such that $x^n = 1$

\item[Dihedral group (2)] $D_{2n} = $ symmetries of an $n$-gon with elements $s$ (flip) and $r$ (rotate). Some properties: Every element can be uniquely written as $s$, and $sr = r\i s$.

\item[Generator, Relation, Presentation (27)] $D_{2n} = \langle r, s | r^n = s^2 = 1, rs = sr\i \rangle$ is a presentation with generators and relations.

\item[Symmetric group $S_n$ (29)] Permutations on $\{1, ..., n\}$. More generally, $S_\Omega$ on an arbitrary set.

\item[Cycle decomposition (29)] Any permutation can be written as disjoint cycles.

\item[Field (34)] A set $F$ with commutative $+, \*$ such that $(F, +), (F-\{0\}, \*)$ are abelian, and the distributive law holds. $F^\times = F - \{0\}$

\item[Distributive law (34)] $a\*(b+c) = a\*b + a\*c$

\item[Quaternion group $Q_8$ (36)] $Q = \{\pm 1, \pm i, \pm j, \pm k\}$. $i^2 = j^2 = k^2 = -1$, $ij=k, jk = i, ki = j$.

\item[Homomorphism (36)] Given: $(G, \star)$ and $(H, \diamond)$, a \emph{homomorphism} is a map $\varphi : G \to H$ such that $\varphi(x \star y) = \varphi(x) \diamond \varphi(y)$.

\item[Isomorphism (37)] $\varphi : G \to H$ is an isomorphism iff $\phi$ is a homomorphism and bijection.

\item[$G \cong H$ (37)] $G$ and $H$ are of the same isomorphism type / there is an isomorphism $\varphi : G \to H$.

\item[Group Action (41)] A (left) $group$ action of a group $G$ on a set $A$ is a map from $G \times A \to A$ (written $g \* a$) such that:

\begin{enumerate}
\item $g_1\*(g_2\*a) = (g_1 g_2)\*a$
\item $1\*a = a$
\end{enumerate}

\item[Permutation Representation (42)] For $g \in G$, we get a map $\sigma_g:A\to A$ with $\sigma_g(a) = g \* a$. $\sigma_g$ is a permutation of $A$. The homomorphism $g \mapsto \sigma_g$ from $G \to S_a$ is the \emph{permutation representation} of the action.

\item[Trivial action (43)] $ga = a$

\item[Faithful action (43)] $g \neq g' \imp ga \neq g'a$ (the associated permutation representation is injective, and the kernel is the identity)

\item[Kernel of an action (43)] $\{g \in G\ |\ \forall b \in B: gb = b\}$

\item[(Left) regular action (44)] With $A = G$, let $g\*a = ga$. This is a group action of $G$ on itself, where (fixed) $g \in G$ permutes the elements of $G$ by (left) multiplication ($g : a \mapsto ga$); left translation if written as $g+a$. This left regular action is faithful.

\item[Subgroup $H \leq G$ (46)] $H \leq G$ iff $H$ is nonempty and closed under products and inverses.

\item[Subgroup criterion (47)] $H \leq G$ iff:

\begin{enumerate}
\item $H \neq \0$
\item $x, y \in H \imp xy\i \in H$
\end{enumerate}

\item[Centralizer $C_G(A)$ (49)] The centralizer of $A$ in $G$ is
\[C_G(A) = \{c \in G\ |\ \forall a \in A: gag\i = a\}\]
Note that $C_G(A) \leq G$\\
The definition could have used $ga = ag$, so this is the set of elements of $G$ that commute with all elements of $A$.

\item[Center $Z(A)$ (50)] $Z(A) = C_G(G)$; the set of elements that commute with all others.

\item[Acting by conjugation $g A g\i$ (50)] $g A g\i = \{gag\i | \in A\}$. Useful for proving that centralizers, normalizers and centers are subgroups.

\item[Normalizer $N_G(A)$ (50)]
\[N_G(A) = \{g \in G | gAg\i = A\}\]
 Note that $C_G(A) \leq N_G(A) \leq G$.

\item[Stabilizer $G_s$ (51)] If $G$ is acting on $S$, then
\[G_s = \{g \in G\ |\ g \* s \}\]
 Note that $G_s \in G$.

\item[Cyclic Group (54)] $H = \{x^n | n \in \Z\}$ generated by $x$

\item[Properties of Cyclic Groups (52-54)] Several intuitive facts about orders of elements hold.

\item[Two cyclic groups of the same order are isomorphic (56)] Theorem. In a cyclic group, subgroups of the same order are the same (not just isomorphic). (57)

\item[Subgroup intersection (62)] The intersection of a nonempty collection of subgroups of $G$ is also a subgroup. (Proof: subgroup criterion)

\item[Subgroup generated by a subset (62)] $\langle A \rangle = \displaystyle \bigcap_{A\subset H \leq G}H$

\item[Word representation (63)] $\left(\langle A \rangle = \right) \ov{A} = \{a_1^{\epsilon_1}a_2^{\epsilon_2}...a_n^{\epsilon_n} | n \in \Z, n \geq 0, a_i \in A, \epsilon = \pm 1\}$ These finite products are are words. Note that if $A$ is abelian, we can collect exponents.

\item[Lattice (66)] The poset of the subgroups of $G$ by the subgroup relation.

\item[Join (67)] $\langle H, K \rangle$, supremum on the lattice.

\item[Meet (not defined/used?)] Infimum on the lattice; intersection.

\item[Kernel of a quotient (75)] If we have a homomorphism $\phi : G \to H$:
\[\ker (\varphi) = \{g \in G | \varphi(g) = 1\}\]
Note that $\ker(\varphi) \leq G$, and $image(\phi) \leq H$.

\item[Quotient group (76)] Take a homomorphism $\varphi: G \to H$ with kernel $K$. $G / K$ (``$G$ mod $K$'') is the group $G$ collapsed into ``fibers'' with representatives in $H$. $G / K$ is isomorphic to the image of $G$ under homomorphism $\phi: G \to H$.

\item[(Left) coset (77)] For any $N \leq G, g \in G$:
\[gN = \{gn\ |\ n \in N\}\]
Right coset: $Ng = \{ng\ |\ n \in N\}$. Note that cosets are elements of the quotient $G / K$ that is isomorphic to the image of $G$.

\item[Representative of a coset (77))] Any element of a coset.

\item[Cosets form a group (77)] If $K$ is the kernel of a homomorphism, $u K \circ v K = (uv)K$ is well-defined.

\item[Cosets as a Partitions (80)] If $N \leq G$, the left cosets of $N$ partition $G$ into equivalence classes. $uN = vN \iff v\i u \in N \iff$ $u$ and $v$ are representatives of the same coset.

\item[Cosets of a subgroup (81)] Let $N \leq G$. The operation on the set of left cosets of $N$ in $G$ described by $uN \* vN = (uv)N$ is well defined iff $g n g\i \in N$ for all $g \in G$, n $\in N$, making them a group. Note that $(g N)\i = g\i N$.

\item[Conjugate of an element/subgroup (82)] $gng\i$ is the conjugate of $n \in N$ by $g$. $gNg\i = \{gng\i | n \in N\}$ is the conjugate of $N$ by $g$. $g$ \emph{normalizes} $N$ iff $g N g\i = N$.

\item[Normal subgroup $N \trianglelefteq G$ (82)] $N$ is normal in $G$ if every $g \in G$ normalizes it. The following are equivalent (Theorem 6):

\begin{enumerate}
\item $N \trianglelefteq G$
\item $N_G(N) = C$
\item $gN = Ng$ for all $g \in G$
\item The operation on left cosets makes it into a group
\item $g N g\i \subseteq N$ for all $g \in G$
\item $N$ is the kernel of some homomorphism of $G$ (Theorem 7)
\end{enumerate}

This is an \emph{embedding property}; it doesn't depend on the structure of $N$, but the relation of $N$ to $G$. (83)

\item[(Natural) Projection Homomorphism (83)] Let $N \trianglelefteq G$. The natural projection (homomorphism) $\pi: G \to G / N$ is:
\[\pi(g) = gN\]

\item[Complete Preimage (83)] If $\ov{H} \leq G/N$ is a subgroup of $G/N$, the complete preimage of $\ov{N}$ in $G$ is the preimage of $\ov{H}$ under the natural projection (it's a subgroup of $G$).

\item[Lagrange's Theorem (89)] If $G$ is a finite group and $H \leq G$, then $|H|$ (the order of $H$) divides $|G|$ and the number of cosets of $H$ in $G$ is ${|G|\/|H|}$.

\item[Index $|G : H|$ (90)] If $G$ is a group and $H \leq G$, the number of left cosets of $H$ in $G$ is the \emph{index} of $H$ in $G$, denoted $|G : H|$.

\item[Simple Group (91)] A group $G$ such that the only normal subgroups are $1$ and $G$.

\item[Cauchy's Theorem (93)] If $G$ is a finite group and a prime $p$ divides $|G|$, then $G$ has an element of order $p^\alpha$. Special case of Sylow.)

\item[Subgroup ``Product''(?) $HK$ (93)] $HK = \{hk | h \in H, k \in K\} = \displaystyle \bigcup_{h \in H} hK$ (union of left cosets).\\
Note that $|HK| = {|H||K|\/|H\cup K|}$ if $H$ and $K$ are finite. Also, if $H$ and $K$ are subgroups, $HK$ is a subgroup iff $HK = KH$.

\item[Corollary 15 (94)] If $H, K \leq G$ and $H \leq N_G{K}$, then $HK$ is a subgroup of $G$.\\
Note that if $K \trianglelefteq G$, then $HK \leq G$ for all $H \leq G$.

\item[Normalizing/Centralizing the kernel (95)] If $A \subseteq N_G(K)$ (or $C_G(K)$), $A$ normalizes $K$ (or centralizes).

\item[First Isomorphism Theorem (97)] If $\phi: G \to H$ (homomorphism), then $\ker \varphi \trianglelefteq G$ and $G / \ker \varphi \cong \varphi(G)$.

\item[Corollaries (97)]

\begin{enumerate}
\item $\varphi$ is injective iff $\ker \phi = 1$.
\item $|G : \ker \varphi| = |\phi(G)|$
\end{enumerate}

\item[Second (Diamond) Isomorphism Theorem (97)] If $A, B \leq G$ and $A \leq N_G(B)$, then:
\begin{enumerate}
\item $AB \leq G$
\item $B \tleq AB$
\item $A \cap B \tleq A$
\item $AB / B \cong A / A \cap B$.
\end{enumerate}

\item[Third Isomorphism Theorem (98)] If $H, K \tleq G$ with $H \leq K$, then:

\begin{itemize}
\item $K / H \tleq G / H$
\item $(G/H)/(K/H) = G/K$
\end{itemize}

\item[Fourth Isomorphism Theorem (99)] If $N \tleq G$, then there is a bijection $f : \{A \leq G | N \subseteq G\} \to \{\ov{A} \in G/N\}$ (the latter elements of the form $A / N$). For all $A, B \leq G$ with $N \leq A, B$:

\begin{enumerate}
\item $A \leq B$ iff $f(A) = f(B)$
\item $A \leq B \imp |B:A| = |f(B) : f(A)|$
\item $f(\langle A, B \rangle) = \langle f(A), f(B) \rangle$
\item $f(A \cap B) = f(A) \cap f(B)$
\item $A \tleq G$ iff $f(A) \tleq f(G)$
\end{enumerate}

\item[Homomorphisms on the quotient group (100)] (defined by representatives) $\phi : G / N \to H$ is well-defined on iff $N \leq \ker \Phi$. ($\Phi : G \to H$)

\item[Transpositions (107)] A transposition is an arbitrary $2$-cycle. Transpositions generate $S_n$.

\item[Sign of a Permutation $\epsilon(\sigma)$ (108)] $\epsilon(\sigma) = \pm 1$ defined in the usual way. Note that $\epsilon: S_n \to \{\pm 1\} \cong \Z / 2\Z$ is a homomorphism. All transpositions are odd permutations, and $\epsilon$ is a surjective homomorphism. (109)

\item[Alternating group $A_n$] $A_n$ is the kernel of the homomorphism $\epsilon$. (i.e. the set of even permutations.)

\item[Action defined by homomorphism] For any group $G$ and non-empty set $A$, there is a bijection between the actions of $G$ on $A$ an the homomorphisms of $G$ into $S_A$.

\item[Permutation Representation (114)] Defined already on page 43. We say $G$ \emph{affords} or \emph{induces} the associated permutation representation of $G$.

\item[Orbits (114)] Let $G$ be group acting on a non-empty set $A$. The following is an equivalence relation:

\[a \sim b \iff \E g \in G: a = g\*b\]

For each $a \in A$, the number of elements in the equivalence class containing $a$ is $G:G_a+$, the index of the stabilizer of $a$.\\

$\{g \* a | g \in G\}$ is called the \emph{orbit} of $G$ containing $a$. (115)

\item[Transitive Action (115)] $G$ acts transitively on $A$ iff there is only one orbit.

\item[Permutation Group (116)] A subgroup of a symmetric group. ``For any subgroup $G$ of $S_n$, the orbits of $G$ will refer to its orbits on $\{1, 2, ..., n\}$. The orbits of an element $\sigma \in S_n$ will mean the orbits of the group $\langle \sigma \rangle$.''

\item[Group acting on itself (118)] $g \* a = ga$ for all $g, a \in G$.

\item[Permutation group of self-action (118)] Label the elements of $G$ (finite) with integers from $S_n$. There are different possible labelings.

\[\sigma_g(i) = j \iff gg_i = g_j\]

\item[Left multiplication on the elements of an action (119)] If $H \leq G$ with $A = $ the cosets of $H \in G$. Define an action of $G$ on $A$ by

\[g \* aH = (ga)H\ \ \ \ (g \in G, aH \in A)\]

Similarly, we can label the cosets $a_1H, a_2H, ..., a_mH$ and define $\sigma_g(i) = ga_iH = a_jH$.

\item[$G$ acting on cosets of $H$ (119)] If $H \leq G$ and $G$ acts by left multiplication on $A$ ($=$ set of left cosets of $H \in G$) with associated permutation representation $\pi_H$.

\begin{enumerate}
\item $G$ acts transitively on $A$
\item The stabilizer in $G$ of the point $1H \in A$ is the (subgroup) $H$
\item The kernel of the action (of $\pi_H$) is $\cap_{x \in G} x H x\i$, and $\ker \pi_H$ is the largest normal subgroup of $G$ contained in $H$.
\end{enumerate}

\item[Cayley's Theorem (120)] Every group is isomorphic to a subgroup of some symmetric group. In particular $G$ is isomorphic to a subgroup of $S_{|G|}$.

\item[Corollary (120)] If $|G|$ is finite, and $p$ is the smallest prime dividing $|G|$, then any subgroup of index $p$ is normal in $G$.

\end{mylist}


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