To receive notifications about scheduled maintenance, please subscribe to the mailing-list gitlab-operations@sympa.ethz.ch. You can subscribe to the mailing-list at https://sympa.ethz.ch

Unverified Commit c0cbd6bf authored by rrueger's avatar rrueger
Browse files

Algebra I: Update definition styling

parent 31621f5d
......@@ -2,7 +2,7 @@
\section{Definition of the Group}
\begin{definition}
\begin{definition}[Group]
A \textbf{group} is a set $G$ equipped with a binary operation $*$ such that
%
\begin{enumerate}
......@@ -24,7 +24,7 @@
for all $x \in G$. The other equations follow from these axioms.
\end{remark}
\begin{definition}
\begin{definition}[Abelian, Commutative]
A group $G$ is called \textbf{abelian} if it satisfies the \textbf{commutative
law}.
\end{definition}
......@@ -50,13 +50,13 @@
\end{enumerate}
\end{proposition}
\begin{definition}
\begin{definition}[Semigroup]
A \textbf{semigroup} is a set having an associative operation; a
\textbf{monoid} is a semigroup $S$ having a (two-sided) identity element 1;
that is, $1s = s= s1$ for all $s \in S$.
\end{definition}
\begin{definition}
\begin{definition}[Order]
Let $G$ be a group and let $a \in G$. If $a^k = 1$ for some $k \geq 1$, then
the smallest such exponent $k \geq 1$ is called the \textbf{order} of $a$;
if no such power exists, then we say that $a$ has \textbf{infinite order}.
......@@ -71,12 +71,12 @@
If $G$ is a finite group, then every $x \in G$ has finite order.
\end{proposition}
\begin{definition}
\begin{definition}[Group order]
If $G$ is a finite group, then the number of elements in $G$, denoted by
$|G|$, is called the \textbf{order} of $G$.
\end{definition}
\begin{definition}
\begin{definition}[Dihedral group]
If $\pi_{n}$ is a regular polygon with $n \geq 3$ vertices $v_1, \ldots, v_n$
and center $O$, then the symmetry group $\Sigma(\pi_n)$ is called the
\textbf{dihedral group} of order $2n$, denoted by $D_{2n}$ or $D_{n}$. The
......@@ -107,7 +107,7 @@
\subsection{Subgroups}
\begin{definition}
\begin{definition}[Subgroup, Proper Subgroup]
A subset $H$ of a group $G$ is a \textbf{subgroup} if
%
\begin{enumerate}
......@@ -135,7 +135,7 @@
group and $\N$ is closed under addition, but $\N$ is not a subgroup of $\Z$.
\end{example}
\begin{definition}
\begin{definition}[Cyclic Group]
If $G$ is a group and $a \in G$, then the \textbf{cyclic subgroup} of $G$
\textbf{generated} by $a$, denoted by $\langle a \rangle$, is
%
......@@ -162,7 +162,7 @@
\cdot)$ for some $m \geq 1$.
\end{lemma}
\begin{definition}
\begin{definition}[Generator]
If $X$ is a subset of a group $G$, then $\langle X \rangle$ is called the
\textbf{subgroup generated by} X.
\end{definition}
......@@ -170,7 +170,7 @@
\subsection{Cosets and Lagrange's Theorem}
\begin{definition}
\begin{definition}[Coset]
If $H$ is a subgroup of a group $G$ and $a \in G$, then the \textbf{(left)
coset} $aH$ is
%
......@@ -199,7 +199,7 @@
\end{enumerate}
\end{lemma}
\begin{definition}
\begin{definition}[Index]
The \textbf{index} of a subgroup $H$ in $G$, denoted by $[G:H]$, is the number
of \emph{left} cosets of $H$ in $G$.
\end{definition}
......@@ -229,7 +229,7 @@
\section{Homomorphisms}
\begin{definition}
\begin{definition}[Homomorphism]
Let $(G,*)$ and $(H,\circ)$ be groups. A \textbf{homomorphism} is a function
satisfying
%
......@@ -248,7 +248,7 @@
\end{enumerate}
\end{lemma}
\begin{definition}
\begin{definition}[Kernel, Image]
If $f \colon G \to H$ is a homomorphism, define the \textbf{kernel} of $f$ by
%
\begin{align*}
......@@ -262,12 +262,12 @@
\end{align*}
\end{definition}
\begin{definition}
\begin{definition}[Conjugate]
Let $G$ be a group. A \textbf{conjugate} of $a \in G$ is an element in $G$ of
the form $gag^{-1}$ for some $g \in G$.
\end{definition}
\begin{definition}
\begin{definition}[Cojugation]
If $G$ is a group and $g \in G$, then \textbf{conjugation by} $g$ is the
function defined by for all $a \in G$
%
......@@ -285,7 +285,7 @@
\end{enumerate}
\end{proposition}
\begin{definition}
\begin{definition}[Normal Subgroup]
A subgroup $K$ of a group $G$ is called a \textbf{normal subgroup} it is
invariant under conjugation from elements in G. That is, $\forall g \in G:
\gamma_g (K) = K$. If $K$ is a normal subgroup of $G$, we write $K
......@@ -313,7 +313,7 @@
\end{enumerate}
\end{proposition}
\begin{definition}
\begin{definition}[Center]
The \textbf{center} of a group $G$, $Z(G)$, is defined by
%
\begin{align*}
......@@ -331,7 +331,7 @@
is normal.
\end{example}
\begin{definition}
\begin{definition}[Inner automorphism, Outer automorphism]
The conjugations are called \textbf{inner automorphisms}. An automorphism that
is not inner is called \textbf{outer automorphism}. The set $\mathrm{Aut}(G)$
of all the automorphisms of $G$ is itself a group. Furthermore,
......@@ -532,7 +532,7 @@
contains a subgroup of order $d$.
\end{proposition*}
\begin{definition*}
\begin{definition*}[Direct Product]
If $H$ and $K$ are groups, then their \textbf{direct product}, denoted by $H
\times K$, is the set of all ordered pairs $(h,k), h \in H, k \in K$, equipped
with the operation
......@@ -567,7 +567,7 @@
\section{Group Actions and the Sylow Theorems}
\begin{definition}
\begin{definition}[Group action]
Let $G$ be a group and $T$ a set. An \textbf{action} of $G$ on $T$ is a map
%
\begin{align*}
......@@ -583,7 +583,7 @@
\end{enumerate}
\end{definition}
\begin{definition}
\begin{definition}[Group of symmetries]
Let $T$ be a set. Then we define
\begin{align*}
\Sym(T) := \set{f \colon T \to T, f \text{~bijective}}
......@@ -618,20 +618,20 @@
functions.
\end{remark}
\begin{definition}
\begin{definition}[Invariant subset, Fixed point]
If $G$ acts on $T$ and $S \subset T$ is such that $g \cdot s \in S, \forall s
\in S$, then $G$ also acts on $S$. We call $S$ a
$\mathbf{G}$\textbf{-invariant subset}. If $S$ consists of a single element
and is invariant, we call that point a \textbf{fixed point} of the action.
\end{definition}
\begin{definition}
\begin{definition}[Faithful group action]
Let $G$ act on $T$. If the homomorphism $\alpha \colon G \to \Sym(T)$ is
injective, then the action is called \textbf{faithful}, i.e. $e_{G} \in G$ is
the only element of $G$ such that $g \cdot t = t, \forall t \in T$.
\end{definition}
\begin{definition}
\begin{definition}[Orbit, Orbit map]
Let $G$ act on $T$. Let $t \in T$. The map $\gamma_t \colon G \to T$ such that
$\gamma_t(g) = g \cdot t$ is called the \textbf{orbit map} associated to $t$.
Its image defines a set called the $\mathbf{G}$\textbf{-orbit} of $t$ in $T$
......@@ -647,7 +647,7 @@
relation. Its equivalence classes are the orbits of $G$ in $T$.
\end{lemma}
\begin{definition}
\begin{definition}[Transitive group action]
Let $G$ act on $T$.
%
\begin{enumerate}
......@@ -657,7 +657,7 @@
\end{enumerate}
\end{definition}
\begin{definition}
\begin{definition}[Stabilizer]
Let $G$ be a group and $T$ a non-empty set with a $G$-action. Let $t \in T$.
We define the \textbf{stabilizer} of $t$ in $G$ to be
%
......@@ -723,13 +723,13 @@
\section{The Symmetric Group}
\begin{definition}
\begin{definition}[Permutation]
A \textbf{permuation} of a set $X$ is a bijection of $X$ to itself. We denote
the family of all permutations of a set $X$ by $S_X$. If $X =
\set{1, 2, \ldots, n}$ we write $S_n$.
\end{definition}
\begin{definition}
\begin{definition}[Cycle]
Let $i_1, i_2, \ldots, i_r$ be distinct integers in $X = \{1, 2, \ldots, n\}$.
If $\alpha \in S_n$ fixes the other integers in $X$ (if any) and if
%
......@@ -750,7 +750,7 @@
A 2-cycle is called \textbf{transposition}.
\end{definition}
\begin{definition}
\begin{definition}[Disjoint permutations]
Two permutations $\alpha, \beta \in S_n$ are \textbf{disjoint} if every $i$
moved by one is fixed by the other.
\end{definition}
......@@ -783,7 +783,7 @@
\end{enumerate}
\end{proposition}
\begin{definition}
\begin{definition}[Complete factorization]
A \textbf{complete factorization} of a permutation $\alpha$ is a factorization
of $\alpha$ into disjoint cycles that contains exactly one 1-cycle $(i)$ for
every $i$ fixed by $\alpha$.
......@@ -795,7 +795,7 @@
in which the cycles occur.
\end{theorem}
\begin{definition}
\begin{definition}[Cycle Structure]
Two permutations have the \textbf{same cycle structure} if, for each $r \geq
1$, their complete factorizations have the same number of r-cycles.
\end{definition}
......@@ -837,13 +837,13 @@
transpositions.
\end{proposition}
\begin{definition}
\begin{definition}[Even permutation, odd permutation]
A permutation $\alpha \in S_n$ is \textbf{even} if it is a product of an even
number of transpositions; $\alpha$ is \textbf{odd} if it is not even. The
\textbf{parity} of a permutations is whether is is even or odd.
\end{definition}
\begin{definition}
\begin{definition}[Sign of permutation]
If $\alpha \in S_n$ and $\alpha = \beta_1 \hdots \beta_t$ is a complete
factorization into disjoint cycles, then \textbf{signum} $\alpha$ is defined
by
......@@ -876,7 +876,7 @@
where $n = |G|$.
\end{lemma}
\begin{definition}
\begin{definition}[Alternating group]
The subset
%
\begin{align*}
......@@ -889,7 +889,7 @@
\section{Simple Groups}
\begin{definition}
\begin{definition}[Simple group]
A group $G$ is called \textbf{simple} if $G \neq \set{1}$ and $G$ has no normal
subgroups other than $\set{1}$ and $G$ itself.
\end{definition}
......
......@@ -35,7 +35,7 @@
\end{enumerate}
\end{proposition}
\begin{definition}
\begin{definition}[Extension Field, finite extension, degree of extension]
If $K$ is a field containing $k$ as a subfield, then $K$ is called an
\textbf{extension field} of $k$, denoted by $K/k$. An extension field $K/k$ is
a \textbf{finite extension} if $K$ is a finite-dimensional vector space over
......@@ -43,7 +43,7 @@
of $K/k$.
\end{definition}
\begin{definition}
\begin{definition}[Algebraic and transcendental elements]
Let $K/k$ be an extension field. An element $\alpha \in K$ is
\textbf{algebraic} over $k$ if there is some nonzero polynomial $f(x) \in
k[x]$ having $\alpha$ as a root; otherwise, $\alpha$ is
......@@ -60,7 +60,7 @@
If $K/k$ is a finite extension field, then $K/k$ is an algebraic extension.
\end{proposition}
\begin{definition}
\begin{definition}[Adjoint]
If $K/k$ is an extension field and $\alpha \in K$, then $k(\alpha)$ is the
intersection of all those subfields of $K$ containing $k$ and $\alpha$; we
call $k(\alpha)$ the subfield of $K$ obtained by \textbf{adjoining} $\alpha$
......@@ -91,7 +91,7 @@
\end{enumerate}
\end{theorem}
\begin{definition}
\begin{definition}[Minimal Polynomial]
If $K/k$ is an extension field and $\alpha \in K$ is algebraic over $k$, then
the unique monic irreducible polynomial $p(x) \in k[x]$ having $\alpha$ as a
root is called the \textbf{minimal polynomial} of $\alpha$ over $k$; it is
......@@ -115,7 +115,7 @@
with $f$ a product of linear polynomials in $K[x]$.
\end{theorem}
\begin{definition}
\begin{definition}[Splitting field]
If $K/k$ is an extension field and $f(X) \in k[x]$ is nonconstant, then $f$
\textbf{splits over} $K$ if $f(x) = a(x-z_1) \hdots (x-z_n)$, where $z_1,
\ldots, z_n$ are in $K$ and $a \in k$. An extension field $E/k$ is called a
......@@ -140,7 +140,7 @@
elements.
\end{theorem}
\begin{definition}
\begin{definition}[Primitive element]
If $k$ is a finite field, a generator of the cyclic group $k^{\times}$ is
called a \textbf{primitive element} of $k$.
\end{definition}
......@@ -202,13 +202,13 @@
\section{Algebraic Closure}
\begin{definition}
\begin{definition}[Algebraially closed]
A field $K$ is \textbf{algebraically closed} if any non-constant $f(x) \in
K[x]$ splits in linear factors over $K$, or equivalently, any non-constant
$f(x) \in K[x]$ has a root in $K$.
\end{definition}
\begin{definition}
\begin{definition}[Algebraic cloure]
Let $K$ be a field. An \textbf{algebraic closure} of $K$ is an algebraic
extension $L/K$ such that $L$ is algebraically closed.
\end{definition}
......@@ -222,8 +222,8 @@
$K$.
\end{theorem}
\begin{definition}
If $F/k$ and $K/k$ are extension fields, then a k-\textbf{map} is a ring
\begin{definition}[$k$-map]
If $F/k$ and $K/k$ are extension fields, then a $k$-\textbf{map} is a ring
homomorphism $\varphi: F \to K$ that fixes $k$ pointwise.
\end{definition}
......
......@@ -9,7 +9,7 @@
\section{The Definition of the Module}
\begin{definition}
\begin{definition}[Left module, Right module]
Let $R$ be a ring. A \textbf{left} $R$\textbf{-module} is an additive abelian
group $M$ equipped with a \textbf{scalar multiplication} $R \times M
\to M$, denoted by
......@@ -56,7 +56,7 @@
\end{remark}
\end{definition}
\begin{definition}
\begin{definition}[$R$-homomorphism]
If $R$ is a ring and $M$ and $N$ are both left $R$-modules (or both right
$R$-modules), then a function $f \colon M \to N$ is an
$R$-\textbf{homomorphism} (or $R$\textbf{-map}) if
......@@ -72,23 +72,23 @@
\section{Fundamental Concepts and Results}
\begin{definition}
Let $M$ be an $R$-module. We define a submodule $N$ of $M$ as a subgroup of
the additive group $(M, +, 0)$ which is closed under the action of the
elements of $R$; that is, if $a \in R$ and $y \in N$, then $ay \in N$.
\begin{definition}[Submodule]
Let $M$ be an $R$-module. We define a \textbf{submodule} $N$ of $M$ as a
subgroup of the additive group $(M, +, 0)$ which is closed under the action of
the elements of $R$; that is, if $a \in R$ and $y \in N$, then $ay \in N$.
\end{definition}
\begin{theorem}
Let $M$ and $M'$ be two $R$-modules and $\phi \colon M \to M'$ an
$R$-homomorphism. Then $\ker \phi \subset M$ and $\im \phi \subset M'$ are
submodules of $M$ and $M'$, respectively. Furthermore, the first isomorphism
theorem also holds for modules, i.e.\
theorem also holds for modules, i.e~
\begin{align*}
M / \ker \phi \cong \im \phi
\end{align*}
\end{theorem}
\begin{definition}
\begin{definition}[Cyclic module]
Let $M$ be an $R$-module. Then $M$ is said to be a \textbf{cyclic module} if
$M = Rx = \{ ax \mid a \in R\}$ where $x \in M$. This is in direct analogy to
a cyclic group, namely they are both generated by one element.
......@@ -99,7 +99,7 @@ we have the homomorphism $\mu_x \colon R \to Rx$ with $r \mapsto rx$.
Evidently, this homomorphism is surjective and we have $M = Rx \cong R/\ker
\mu_x$. This motivates the following definition.
\begin{definition}
\begin{definition}[Annihilator]
Let $M = Rx$ be a cyclic $R$-module. Let $\mu_x \colon R \to Rx$ be defined by
$r \mapsto rx$. Then we define the \textbf{annihilator} of $\mu_x$
%
......@@ -130,7 +130,7 @@ Evidently, this homomorphism is surjective and we have $M = Rx \cong R/\ker
\section{Direct Sums of Modules}
\begin{definition}
\begin{definition}[Direct sum of modules]
Let $M_1, \dotsc, M_n$ be modules over the same ring $R$. Let $M$ be the
product set $M_ 1 \times \dotsm \times M_n$. In the special case of the free
module we define addition, the zero element and multiplication by elements in
......@@ -193,7 +193,7 @@ $K$. Then we have $M \cong D^{(n)}/K$. The core idea is now to examine $K$.
elements.
\end{theorem}
\begin{definition}
\begin{definition}[Equivalence of Matrices]
Let $D$ be a PID\@. Two matrices $A, B \in M_{m \times n}(D)$ are said to be
\textbf{equivalent} if there exist invertible matrices $P \in M_{n}(D), Q \in
M_{n}(D)$, such that $B = PAQ$.
......@@ -228,7 +228,7 @@ $K$. Then we have $M \cong D^{(n)}/K$. The core idea is now to examine $K$.
This decomposition is in general not unique!
\end{remark}
\begin{definition}
\begin{definition}[Torsion module]
Let $M$ be a finitely generated module over a PID and define the
\textbf{torsion module} of $M$, $\tor M$, by
%
......@@ -245,7 +245,7 @@ $K$. Then we have $M \cong D^{(n)}/K$. The core idea is now to examine $K$.
and a free module.
\end{theorem}
\begin{definition}
\begin{definition}[$p$-component]
If $p$ is a prime, we define the \textbf{p-component} $M_p$ of $M$ by
\begin{align*}
M_p = \set{y \in M \mid p^k y = 0 \text{~for some~} k \in \N}.
......@@ -265,7 +265,7 @@ $K$. Then we have $M \cong D^{(n)}/K$. The core idea is now to examine $K$.
= t$ and $\ann z_i = \ann w_i$ for all $1 \leq i \leq s$.
\end{theorem}
\begin{definition}
\begin{definition}[Finitely generated]
An abelian Group $G$ is called \textbf{finitely generated} if there exist
finitely many elements $x_1, \dotsc, x_s \in G$ such that every $x$ in $G$ can
be written in the form
......
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment