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 19
 20\title{Notes on "A book of Abstract Algebra", Charles C. Pinter}
 21\author{arnaucube}
 22\date{February 2022}
 23
 24\begin{document}
 25
 26\maketitle
 27
 28\begin{abstract}
 29    Notes on \emph{"A book of Abstract Algebra - by Charles C. Pinter"}, is a $LaTeX$ version of handmade notes taken while reading the book. It contains only some definitions and theorems (without proofs), so it is highly recommended to read the actual book instead of the current notes. Additionally, some theorems and concepts are extended with notes from other resources from outside the book.
 30    \\
 31    \emph{This is an unfinished and 'work in progress' document.}
 32\end{abstract}
 33
 34\tableofcontents
 35% \newpage
 36
 37% TODO maybe this 'Introduction' section should be removed
 38% \section{Introduction}
 39% 
 40% $\mathbb{R} \supseteq \mathbb{Q} \supseteq \mathbb{Z} \supseteq \mathbb{N}$
 41% \begin{itemize}
 42%     \item $\mathbb{R}$ (reals): $\{ \ldots, -3, 0, \sqrt{2}, 2, e, \pi, 4, \ldots \}$
 43%     \item $\mathbb{Q}$ (rationals): $\{ \frac{a}{b} \mid a, b \in \mathbb{Z} \wedge b \neq 0 \}$
 44%     \item $\mathbb{Z}$ (integers): $\{ \ldots, -3, -1, 0, 1, 3, \ldots \}$
 45%     \item $\mathbb{N}$ (naturals): non negative integers, $\{1, 2, 3, ...\}$
 46% \end{itemize}
 47
 48\section{Groups}
 49
 50\begin{definition}[Group]
 51    A set $G$ with an operation $*$ which satisfies the axioms:
 52    \begin{enumerate}[i.]
 53	\item $*$ is \emph{associative}
 54	\item \emph{(identity element)} there is an element $e \in G$ s.t. $a * e = a$ and $e * a = a$ $\forall a \in G$
 55	\item \emph{(inverse)} $\forall a \in G$, there is an element $a^{-1} \in G$ s.t. $a*a^{-1} = e$ and $a^{-1} * a =e$
 56    \end{enumerate}
 57\end{definition}
 58
 59\begin{definition}[Abelian Group]
 60    A group $G$ is said to be \emph{commutative} if $\forall a, b \in G$, $ab = ba$. A commutative group is also called \emph{Abelian}.
 61\end{definition}
 62
 63\begin{definition}[Order of an element]
 64    In a group $G$, the order of an element $a \in G$ is the least positive integer $n$ such that $a \cdot a \cdots a = a^n = e$. It is represented by $ord(a)$.
 65\end{definition}
 66
 67\begin{definition}[Order of a group]
 68    Order of a group $G$, is the number of elements in $G$. It is represented by $|G|$.
 69\end{definition}
 70
 71\begin{definition}[Cyclic group]
 72    Let $G$ be a group, and $a \in G$. If $G$ consists of all the powers of $a$ and nothing else:
 73    $$G = \{a^n : n \in \mathbb{Z}\}$$
 74    then, $G$ is called a \emph{cyclic group}, and $a$ is called its \emph{generator}.
 75    \\
 76    The group $G$ generated by $a$ is defined by $G=\langle a \rangle$.
 77\end{definition}
 78
 79\begin{theorem}
 80    The \emph{order of a cyclic group} is the same as the \emph{order of it's generator}. In other words, for a cyclic group, $|\langle a \rangle | = ord(a)$.
 81    \begin{itemize}
 82	\item[] $\langle a \rangle$ defines a cyclic group generated by $a$. $\langle a \rangle = \{e, a, a^2, ..., a^{n-1}\}$
 83	\item[] $| \langle a \rangle |$ defines the order of the cyclic group generated by $a$.
 84    \end{itemize}
 85\end{theorem}
 86
 87\begin{theorem}
 88Every subgroup of a cyclic group is cyclic.
 89\end{theorem}
 90
 91
 92
 93\section{Subgroups}
 94\begin{definition}[Subgroup]
 95    Let $G$ be a group, and $H$ a non-empty subset of $G$. If
 96    \begin{enumerate}[i.]
 97	\item the idenity $e$ of $G$ is in $H$.
 98	\item $H$ is closed with respect to the operation. Which is for $a, b \in H$, $ab \in H$.
 99	\item $H$ is closed with respect to inverses. Which is for $a \in H$, $a^{-1} \in H$.
100    \end{enumerate}
101    we call $H$ a \emph{subgroup} of $G$. The operation of $H$ is the same as the operation of $G$.
102\end{definition}
103
104\begin{theorem}
105    Every subgroup of a cyclic group is cyclic.
106\end{theorem}
107
108\section{Functions}
109\begin{definition}[Function]
110    If $A$ and $B$ are sets, then a function from $A$ to $B$ is a rule which to every element $x$ in $A$ assigns a unique element $y$ in $B$.
111    \\
112    Functions are represented by $f: A \rightarrow B$, where $\forall a \in A \Rightarrow f(a) \in B$.
113\end{definition}
114
115\begin{definition}[Injective (monomorphism)]
116    A function $f:A \rightarrow B$ is called \emph{injective} if each element of $B$ is the image of no more than one element of $A$.
117\end{definition}
118
119\begin{definition}[Surjective (epimorphism)]
120    A function $f:A \rightarrow B$ is called \emph{surjective} if each element of $B$ is the image of at least one element of $A$.
121\end{definition}
122
123\begin{definition}[Bijective (isomorphism)]
124    A function $f:A \rightarrow B$ is called \emph{bijective} if it is both \emph{injective} and \emph{surjective}.
125    \\
126    A function $f: A \rightarrow B$ has an inverse iff it is \emph{bijective}. In that case, the inverse $f^{-1}$ is a bijective function from $B$ to $A$.
127\end{definition}
128
129\begin{definition}[Composite function]
130    A function $f:A \rightarrow B$ and $g: B \rightarrow C$ be functions. The \emph{composite function} denoted by $g \circ f$ is a function from $A$ to $C$ defined as follows:
131    $$[g \circ f](x) = g(f(x)),  \forall x \in A$$
132\end{definition}
133
134
135\begin{definition}[Permutation]
136    By a \emph{permutation} of a set $A$ we mean a \emph{bijective function from $A$ to $A$}, that is, a one-to-one correspondence between $A$ and itself.
137    \\
138    The set of all the permutations of $A$, with the operation $\circ$ of composition, is a group.
139    \\
140    For any positive integer $n$, the symmetric group on the set ${1,2, 3,..., n}$ is called the \emph{symmetric group on $n$ elements}, and is denoted by $S_n$.
141\end{definition}
142
143% TODO define 'group of permutations' & 'cycle of permutations'
144
145\section{Isomorphism}
146
147\begin{definition}[Isomorphism]
148    Let $G_1$ and $G_2$ be groups. A bijective function $f: G_1 \rightarrow G_2$ with the property that for any two elements $a, b \in G_1$,
149    $$f(ab) = f(a)f(b)$$
150    is called an \emph{isomorphism} from $G_1$ to $G_2$.
151    \\
152    If there exists an isomorphism from $G_1$ to $G_2$, we say that $G_1$ is \emph{isomorphic} to $G_2$, symbolized by $G_1 \cong G_2$.
153\end{definition}
154
155
156\begin{theorem}[Cayley's Theorem]
157    Every group is isomorphic to a group of permutations.
158\end{theorem}
159
160\begin{theorem}
161    (Isomorphism of cyclic groups)
162    \begin{enumerate}[i.]
163	\item For every positive integer $n$, every cyclic group of order $n$ is isomorphic to $\mathbb{Z}_n$. Thus, any two cyclic groups of order $n$ are isomorphic to each other.
164	\item Every cyclic group of order infinity is isomorphic to $\mathbb{Z}$, and therefore any two cyclic groups of order infinity are isomorphic to each other.
165    \end{enumerate}
166\end{theorem}
167
168\section{Cosets}
169
170\begin{definition}[Coset]
171    Let $G$ be a group, and $H$ a subgroup of $G$. For any element $a$ in $G$, the symbol $aH$ denotes the set of all products $ah$, as $a$ remains fixed and $h$ ranges over $H$. $aH$ is caled a \emph{left coset} of $H$ in $G$.
172    \\
173    In similar fashion, $Ha$ denotes the set of all products $ha$, as $a$ remains fixed an $h$ ranges over $H$. $Ha$ is called a \emph{right coset} of $H$ in $G$.
174\end{definition}
175
176\begin{theorem}
177    If $Ha$ is any coset of $H$, there is a one-to-one correspondence from $H$ to $Ha$ (there is a bijection between $H$ and $Ha$).\\
178    If $a \in G$, then $|H| = |Ha|$.
179\end{theorem}
180
181\begin{theorem}[Lagrange's theorem]
182    Let $G$ be a finite group, and $H$ any subgroup of $G$. The order of $G$ is a multiple of the order of $H$.
183\end{theorem}
184    Lagrange's theorem can be easily seen by the facts that:
185    \begin{enumerate}[i.]
186	\item cosets partition the group G
187	\item $|Ha| = |H|$ (each coset has the same order as H). 
188    \end{enumerate}
189
190
191By consequence,
192\begin{theorem}
193    If $G$ is a group with a prime number $p$ of elements, then $G$ is a cyclic group. Furthermore, any element $a \neq e$ in $G$ is a generator of $G$.
194\end{theorem}
195
196Thus,
197\begin{theorem}
198    The order of any element of a finite group divides the order of the group.
199\end{theorem}
200
201\begin{definition}[Index of H in G]
202    Number of cosets of H in G. Represented by $(G:H)$.\\
203    Combined with \emph{Lagrange Theorem}, we know that $|G| = |H| \cdot |G:H|$, so,
204    $$(G:H) = \frac{|G|}{|H|}$$
205\end{definition}
206
207
208\section{Homomorphisms}
209
210\begin{definition}[Homomorhism]
211    If $G$ and $G$ are groups, a \emph{homomorphism} from $G$ to $H$ is a function $f: G \rightarrow H$ s.t. for any two elements $a, b \in G$,
212    $$f(ab) = f(a)f(b)$$
213    If there exists a homomorphism from $G$ \emph{onto} $H$, we say that $H$ is a \emph{homomorphic image} of $G$.
214\end{definition}
215Note: an \emph{isomorphism} is a \emph{bijective} \emph{homomorphism}.
216\\
217Example of an \emph{homomorphism}: $f: \mathbb{Z}_6 \rightarrow \mathbb{Z}_3$.
218
219
220\begin{theorem}
221    Let $G$ and $G$ be groups, and $f: G \rightarrow H$ a homomorphism. Then
222    \begin{enumerate}[i.]
223	\item $f(e) = e$
224	\item $f(a^{-1}) = [f(a)]^{-1}, \quad \forall a \in G$
225    \end{enumerate}
226\end{theorem}
227
228\begin{definition}[Conjugate]
229    A \emph{conjugate} of $a$ is any element of the form $xax^{-1}$, where $x \in G$.
230\end{definition}
231
232\begin{definition}[Normal subgroup]
233    Let $H$ be a subgroup of a group $G$. $H$ is called a \emph{normal} subgroup of $G$ if it is closed with respect to conjugates, that is, if\\
234    for any $a \in H$ and $x \in G$, $xax^{-1} \in H$.
235    \\
236    Alternatively, we can see that $H$ is a \emph{normal} subgroup iff $\forall a \in G, aH = Ha$.
237    \\
238    In an abelian group, every subgroup is normal.
239\end{definition}
240
241\begin{definition}[Kernel]
242    Let $f: G \rightarrow H$ be a homomorphism. The \emph{kernel} of $f$ is the set $K$ of all the elements of $G$ which are carried by $f$ onto the neutral element of $H$. That is,
243    $$K = {x \in G : f(x) = e}$$
244\end{definition}
245For every homomorphism, the $e \in G$ maps to $e \in H$, so the \emph{kernel} is never empty, it always contains the identity $e_G$, and if the kernel only contains the identity, then $f$ is one-to-one (injective).
246
247
248\section{Quotient Groups}
249
250Quotient group construction is useful as a way of actually manufacturing all the homomorphic images of any group G. Additionally, we can often choose $H$ so as to "factor out" unwanted properties of $G$, and prserve in $G/H$ only "desirable" traits.
251
252\begin{definition}[Coset multiplication]
253    The coset of $a$, multiplied by the coset of $b$, is defined to be the coset of $ab$. In symbols, $Ha \cdot Hb = H(ab)$.
254\end{definition}
255
256\begin{theorem}
257    Let $H$ be a normal subgroup of $G$. If $Ha = Hc$ and $Hb = Hd$, then $H(ab) = H(cd)$.
258\end{theorem}
259
260\begin{definition}
261    $G/H$ denotes the set which consists of \emph{all the cosets of $H$}.
262    \\
263    Thus, if $Ha, Hb, Hc, \ldots$ are cosets of $H$, then $G/H = \{ Ha, Hb, Hc, ... \}$.
264\end{definition}
265
266\begin{theorem}[Quotient group]
267    $G/H$ with coset multiplication is a group.
268\end{theorem}
269
270\begin{theorem}
271    $G/H$ is a homomorphic image of G.
272    \\
273    Conversely, every homomorphic image of $G$ is a quotient group of $G$.
274\end{theorem}
275
276\begin{theorem}
277    Let $G$ be a group and $H$ a subgroup of $G$. Then
278    \begin{enumerate}[i.]
279	\item $Ha = Hb$ iff ${ab}^{-1} \in H$
280	\item $Ha = H$ iff $a \in H$
281    \end{enumerate}
282\end{theorem}
283
284
285\section{Rings}
286
287\begin{definition}[Ring]
288    A set $A$ with operations called \emph{addition} and \emph{multiplication} which satisfy the following axions:
289    \begin{enumerate}[i.]
290	\item $A$ with addition alone is an abelian group.
291	\item Multiplication is associative.
292	\item Multiplication is distributive over addition. That is, $\forall a,b,c \in A$,
293	    $$a(b+c) = ab + ac$$
294	    $$(b+c)a = ba + ca$$
295    \end{enumerate}
296\end{definition}
297
298\begin{definition}[Commutative ring]
299    By definition, addition is commutative in every ring but multiplication is not. When multiplication also is commutative in a ring, we call that ring a \emph{commutative} ring.
300\end{definition}
301
302\begin{definition}[Unity]
303    A ring does not necessarily have a neutral element for multiplication. If there is in $A$ a neutral element for mulitplication, it is called the \emph{unity} of $A$, and is denoted by the symbol $1$.
304    \\
305    If $A$ has a unity, we call $A$ a \emph{ring with unity}.
306\end{definition}
307
308\begin{definition}[Field]
309    If $A$ is a commutative ring with unity in which every nonzero element is invertible, $A$ is called a \emph{field}.
310\end{definition}
311
312\begin{definition}[Divisor of zero]
313    In any ring, a nonzero element a is called a \emph{divisor of zero} if there is a
314    nonzero element b in the ring such that the product ab or ba is equal to
315    zero.
316\end{definition}
317
318\begin{definition}[Cancellation property]
319A ring is said to have the cancellation property if $ab = ac$ or $ba = ca$ implies $b = c$
320for any elements a, b, and c in the ring if $a \neq 0$.
321\end{definition}
322
323\begin{theorem}
324    A ring has the \emph{cancellation property} iff it has no \emph{divisors of zero}.
325\end{theorem}
326
327\begin{definition}[Integral domain]
328    An \emph{integral domain} is defined to be a commutative ring with unity having the cancellation property.
329\end{definition}
330
331Every field is an integral domain, but the converse is not true (eg. $\mathbb{Z}$ is an integral domain but not a field).
332
333\begin{definition}[Ideal]
334    A nonempty subset $B$ of a ring $A$ is called an \emph{ideal} of $A$ if $B$ is closed with respect to addition and negatives, and $B$ absorbs products in $A$.
335\end{definition}
336
337\framebox{WIP: covered until chapter 18, work in progress.}
338
339\end{document}