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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.
 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 $H$. $|H|$ divides $|G|$.
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
190By consequence,
191\begin{theorem}
192    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 $G$.
193\end{theorem}
194
195Thus,
196\begin{theorem}
197    The order of any element of a finite group divides the order of the group.
198\end{theorem}
199
200\begin{definition}[Index of H in G]
201    Number of cosets of H in G. Represented by $(G:H)$.\\
202    Combined with \emph{Lagrange Theorem}, we know that $|G| = |H| \cdot |G:H|$, so,
203    $$(G:H) = \frac{|G|}{|H|}$$
204\end{definition}
205
206
207\section{Homomorphisms}
208
209\begin{definition}[Homomorhism]
210    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 $a, b \in G$,
211    $$f(ab) = f(a)f(b)$$
212    If there exists a homomorphism from $G$ \emph{onto} $H$, we say that $H$ is a \emph{homomorphic image} of $G$.
213\end{definition}
214Note: an \emph{isomorphism} is a \emph{bijective} \emph{homomorphism}.
215\\
216Example of an \emph{homomorphism}: $f: \mathbb{Z}_6 \rightarrow \mathbb{Z}_3$.
217
218
219\begin{theorem}
220    Let $G$ and $G$ be groups, and $f: G \rightarrow H$ a homomorphism. Then
221    \begin{enumerate}[i.]
222	\item $f(e) = e$
223	\item $f(a^{-1}) = [f(a)]^{-1}, \quad \forall a \in G$
224    \end{enumerate}
225\end{theorem}
226
227\begin{definition}[Conjugate]
228    A \emph{conjugate} of $a$ is any element of the form $xax^{-1}$, where $x \in G$.
229\end{definition}
230
231\begin{definition}[Normal subgroup]
232    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\\
233    for any $a \in H$ and $x \in G$, $xax^{-1} \in H$.
234    \\
235    Alternatively, we can see that $H$ is a \emph{normal} subgroup iff $\forall a \in G, aH = Ha$.
236    \\
237    In an abelian group, every subgroup is normal.
238\end{definition}
239
240\begin{definition}[Kernel]
241    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,
242    $$K = {x \in G : f(x) = e}$$
243\end{definition}
244For 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).
245
246
247\section{Quotient Groups}
248
249Quotient 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.
250
251\begin{definition}[Coset multiplication]
252    The coset of $a$, multiplied by the coset of $b$, is defined to be the coset of $ab$. In symbols, $Ha \cdot Hb = H(ab)$.
253\end{definition}
254
255\begin{theorem}
256    Let $H$ be a normal subgroup of $G$. If $Ha = Hc$ and $Hb = Hd$, then $H(ab) = H(cd)$.
257\end{theorem}
258
259\begin{definition}
260    $G/H$ denotes the set which consists of \emph{all the cosets of $H$}.
261    \\
262    Thus, if $Ha, Hb, Hc, \ldots$ are cosets of $H$, then $G/H = \{ Ha, Hb, Hc, ... \}$.
263\end{definition}
264
265\begin{theorem}[Quotient group]
266    $G/H$ with coset multiplication is a group.
267\end{theorem}
268
269\begin{theorem}
270    $G/H$ is a homomorphic image of G.
271    \\
272    Conversely, every homomorphic image of $G$ is a quotient group of $G$.
273\end{theorem}
274
275\begin{theorem}
276    Let $G$ be a group and $H$ a subgroup of $G$. Then
277    \begin{enumerate}[i.]
278	\item $Ha = Hb$ iff ${ab}^{-1} \in H$
279	\item $Ha = H$ iff $a \in H$
280    \end{enumerate}
281\end{theorem}
282
283
284\section{Rings}
285
286\begin{definition}[Ring]
287    A set $A$ with operations called \emph{addition} and \emph{multiplication} which satisfy the following axioms:
288    \begin{enumerate}[i.]
289	\item $A$ with addition alone is an abelian group.
290	\item Multiplication is associative.
291	\item Multiplication is distributive over addition. That is, $\forall a,b,c \in A$,
292	    $$a(b+c) = ab + ac$$
293	    $$(b+c)a = ba + ca$$
294    \end{enumerate}
295\end{definition}
296
297\begin{definition}[Commutative ring]
298    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.
299\end{definition}
300
301\begin{definition}[Unity]
302    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 $1$.
303    \\
304    If $A$ has a unity, we call $A$ a \emph{ring with unity}.
305\end{definition}
306
307\begin{definition}[Field]
308    If $A$ is a commutative ring with unity in which every nonzero element is invertible, $A$ is called a \emph{field}.
309\end{definition}
310
311\begin{definition}[Divisor of zero]
312    In any ring, a nonzero element a is called a \emph{divisor of zero} if there is a
313    nonzero element b in the ring such that the product ab or ba is equal to
314    zero.
315\end{definition}
316
317\begin{definition}[Cancellation property]
318A ring is said to have the cancellation property if $ab = ac$ or $ba = ca$ implies $b = c$
319for any elements a, b, and c in the ring if $a \neq 0$.
320\end{definition}
321
322\begin{theorem}
323    A ring has the \emph{cancellation property} iff it has no \emph{divisors of zero}.
324\end{theorem}
325
326\begin{definition}[Ideal]
327    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 $A$.
328    \\
329    (\emph{Absorbs product}: $\forall b \in B$ and $x \in A$, then $xb, bx \in B$).
330\end{definition}
331
332\begin{definition}[Principal ideal]
333    A \emph{principal ideal} is an ideal $I$ in a ring $R$ that is generated by a single element $a \in R$ through multiplication by every element of $R$. In other words $I = aR = \{a r : r \in R \}$.
334    \\
335    (eg. Every ideal of $\mathbb{Z}$ is principal).
336\end{definition}
337
338\begin{definition}[Integral domain]
339    An \emph{integral domain} is defined to be a commutative ring with unity having the cancellation property.
340\end{definition}
341
342Every field is an integral domain, but the converse is not true (eg. $\mathbb{Z}$ is an integral domain but not a field).
343
344\begin{definition}[Characteristic n]
345    Let $A$ be a ring with unity, the \emph{characteristic} of $A$ is the least positive integer $n$ such that
346    $$1 + 1 + \cdots + 1 = 0$$
347    If there is no such positive integer $n$, $A$ has characteristic $0$.
348\end{definition}
349
350\section{Elements of number theory}
351
352\begin{definition}[Euclid's lemma]
353    Let $m$ and $n$ be integers, and let $p$ be a prime. If $p|(mn)$, then either $p|m$ or $p|n$.
354\end{definition}
355
356\begin{theorem}[Factorization into primes]
357    Ever integer $n>1$ can be expressed as a product of positive primes. That is, there are one or more primes $p_1, \ldots, p_r$ such that $n=p_1 p_2 \cdots p_r$.
358\end{theorem}
359
360\begin{theorem}[Unique factorization]
361    Suppose $n$ can be factored into positive primes in two ways, namely,
362    $$n= p_1 \cdots p_r = q_1 \cdots q_t$$
363    Then $r=t$, and the $p_i$ are the same numbers as the $q_j$ except, possibly, for the order in which they appear.
364\end{theorem}
365
366From the last two theorems: every integer $m$ can be factored into primes, and the prime factors of $m$ are unique (except for the order).
367
368\begin{theorem}[Little theorem of Fermat]
369    Let $p$ be a prime. Then,
370    $$a^{p-1} \equiv 1 \pmod p, \forall a \not\equiv 0 \pmod p$$
371    \\
372    So, by taking $a^{p-2} \cdot a \equiv 1 \pmod p$, where $a^{p-2} \equiv a^{-1} \pmod p$ (the inverse modulo p), we see that $a^p \equiv a \pmod p, \forall a \in \mathbb{Z}$, so $a^p - a$ is a multiple of $p$.
373
374    ~\\\emph{Relation to Lagrange's theorem:}\\
375    Let $G = \mathbb{Z}_p$, and let $H$ be the multiplicative subgroup of $G$ generated by $a$ (ie. $H = \{ 1, a, a^2, \ldots \}$). The order of $H$ ($h = |H|$), is also the order of $a$ (ie. smallest $n>1$ s.t. $a^n=1~mod~p$).
376
377    By Lagrange's theorem, $h~|~|G| = p - 1$, so $p-1 = h \cdot m$, thus
378    $$
379    a^{p-1} = (a^h)^m \equiv 1^m \equiv 1~mod~p
380    $$
381
382    ~\\\emph{Another perspective:}\\
383    We have $a^p \equiv a \pmod{p}$, by dividing by $a$ on both sides, we obtain $a^{p-1} \equiv 1 \pmod{p}$.
384
385\end{theorem}
386
387\begin{theorem}[Euler's $\phi$ function]
388    \emph{Euler's $\phi$ function} describes the number of integers in $\mathbb{Z} / n \mathbb{Z}$ which are relatively prime (coprime) to $n$.
389\end{theorem}
390
391\begin{theorem}[Euler's theorem]
392    If $a$ and $n$ are relatively prime, $$a^{\phi(n)} \equiv 1 \pmod n$$
393\end{theorem}
394
395
396\section{Polynomials}
397
398\begin{definition}
399    Let $A$ be a commutative ring with unity, and $x$ an arbitrary symbol. Every expression of the form
400    $$a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$$
401    is called a \emph{polynomial in $x$ with coefficients in $A$}, or more simply, a \emph{polynomial in $x$ over $A$}.
402\end{definition}
403 The expressions $a_k x^k$, for $k \in \{ 1, \ldots, n \}$, are called the \emph{terms} of the polynomial, being $a_n x^n$ the \emph{leading term}, and $a_0$ the \emph{constant term}.
404 The $a_k$ are called the \emph{coefficients} of $x^k$, being $a_n$ the \emph{leading coefficient}. And the \emph{degree} of a polynomial $a(x)$ is the greatest $n$ such that the coefficient of $x^n$ is not zero.
405 The polynomial whose leading coefficient is equal to $1$ is called \emph{monic}.
406
407\begin{theorem}[Division algorithm for polynomials]
408    If $a(x)$ and $b(x)$ are polynomials over a field $F$, and $b(x) \neq 0$, there exist polynomials $q(x)$ and $r(x)$ over $F$ such that
409    $a(x) = b(x) q(x) + r(x)$ and
410    [$r(x) = 0$ or $\deg r(x) < \deg b(x)$].
411\end{theorem}
412
413\begin{theorem}
414    Any two nonzero polynomials $a(x), b(x) \in F[x]$ have a $\gcd d(x)$. Furthermore, $d(x)$ can be expressed as a \emph{linear combination}
415    $$d(x)= r(x) a(x) + s(x) b(x)$$
416    where $r(x), s(x) \in F[x]$.
417\end{theorem}
418
419\begin{theorem}[Factorization into irreducible polynomials]
420    Every polynomial $a(x)$ of positive degree in $F[x]$ can be written as a product
421    $$a(x) = k p_1(x) p_2(x) \cdots p_r(x)$$
422    where $k$ is a constant in $F$ and $p_1(x), \ldots, p_r(x)$ are monic irreducible polynomials of $F[x]$.
423\end{theorem}
424
425\begin{theorem}[Unique factorization]
426    If $a(x)$ can be written in two ways as a product of monic irreducibles, say
427    $$a(x) = k p_1(x) \cdots p_r(x) = l q_1(x) \cdots q_s(x)$$
428    then $k=l$, $r=s$, and $p_i(x) = q_j(x)$.
429\end{theorem}
430
431\begin{theorem}
432    $c$ is a root of $a(x)$ iff $x - c$ is a factor of $a(x)$.
433\end{theorem}
434\begin{theorem}
435    If $a(x)$ has distinct roots $c_1, \ldots, c_m$ in $F$, then $(x-c_1)(x-c_2)\cdots(x-c_m)$ is a factor of $a(x)$.
436\end{theorem}
437
438\begin{theorem}
439    If $a(x)$ has degree $n$, it has at most $n$ roots.
440\end{theorem}
441
442
443\framebox{WIP: covered until chapter 26, work in progress.}
444
445\end{document}