Definition Index
This page lists all mathematical definitions found across the site. Click on any definition to jump to its location in the notes.
Abelian groups are groups whose operation is commutative. For $a,b \in G, a * b = b * g$.
The subgroup of $S_n$ consisting of all even permutations of $n$ letters is the alternating group $A_n$ on $n$ letters. If $n \geq 2$, then this set forms a subgroup of $S_n$ of order $n!/2$.
A set $A$ is said to be at most countable if $A$ is finite or countable.
Given $\vec{x} \in \mathbb{R}^k, r > 0,$ the open or closed ball with center $\vec{x}$ and radius $r$ is defined as the set of points $\vec{y}$ such that $|\vec{x} - \vec{y}| < r$ or $|\vec{x} - \vec{y}| \leq r,$ respectively.
$E$ is bounded if there is a real number $M$ and a point $q \in X$ such that $d(p, q) < M$ for all $p \in E.$
Let $E_0$ be the interval $[0, 1].$ Remove the segment $(\frac{1}{3}, \frac{2}{3}),$ and let
$$ E_1 = [0, \frac{1}{3}] \cup [\frac{2}{3}, 1]. $$
Similarly, remove the middle thirds of these intervals, and let
$$E_2 = [0, \frac{1}{9}] \cup [\frac{2}{9}, \frac{3}{9}] \cup [\frac{6}{9}, \frac{7}{9}] \cup [\frac{8}{9}, 1]. $$
We can continue this forever, and we get a nested sequence $\{E_n\}$ of compact sets $E_n$ where:
(a) $E_{n+1} \subset E_n.$
(b) $E_n$ is the union of $2^n$ intervals, each of length $1/3^n.$
Finally, the set
$$ P = \bigcap_{n=1}^\infty E_n $$
is called the Cantor set.
Given two sets, $A$ and $B$, if there is a bijection (a one-to-one mapping of $A$ onto $B$) between $A$ and $B$, we say $A$ and $B$ have the same cardinal number, or that $A$ and $B$ are equivalent. We denote this as $A \sim B$.
The center of a group $G$ is all the elements that commute with all elements of $G$:
$$ Z(G) = \{ z \in G | zg = gz \text{ for all } g \in G \}. $$
$E$ is closed if every limit point of $E$ is a point of $E.$
If $X$ is a metric space, $E \subset X,$ and $E'$ denotes the set of all limit points of $E$ in $X,$ then the closure of $E$ is the set $\overline{E} = E \cup E'.$
A ring in which multiplication is commutative is called a commutative ring.
The commutator subgroup of $G$ is the group $C$ generated by all elements of the set
$$ \{aba^{-1}b^{-1} | a,b \in G\}. $$
A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover. More explicitly, the requirement is that if $\{G_\alpha\}$ is an open cover of $K,$ then there are finitely many indicies $\alpha_1, \dots, \alpha_n$ such that
$$ K \subset G_{\alpha_1} \cup \cdots \cup G_{\alpha_n}. $$
The complement of $E$ (denoted by $E^c$) is the set of all points $p \in X$ such that $p \notin E.$
If $X$ is a metric space, a set $E \subset X$ is said to be connected if $E$ is not a union of two nonempty separated sets.
A set $S$ is said to be connected if every pair of points in $S$ can be joined by a finite number of line segments joined end to end that lie entirely within $S$.
A set $E \subset \mathbb{R}^k$ is said to be convex if
$$ \lambda \vec{x} + (1 - \lambda)\vec{y} \in E $$
whenever $\vec{x}, \vec{y} \in E,$ and $0 < \lambda < 1.$
In geometric terms, this means a set is convex if we can connect any two points in the set with a line segment whose points are all within the set.
Let $H$ be a subgroup of $G$. Given $a \in G$, the subset $aH = \{ah | h \in H\}$ of $G$ is the left coset of $H$ containing $a$, while the subset $Ha = \{ha | h \in H\}$ is the right coset of $H$ containing $a$.
A set $A$ is said to be countable if there exists a bijection between $A$ and the set of all positive integers $\mathbb{Z}_{>0}$, that is, if $A \sim \mathbb{Z}_{>0}.$
A cyclic group is a group where there exists some $g \in G$ such that every element in $G$ can be generated from the group operation applied to $g$.
That is, $G = \{g^N | n \in \mathbb{Z}\}$ when we think of the operation as multiplication, or $G = \{ng | n \in \mathbb{Z}\}$ when we think of the operation as addition.
Any subgroup of a cyclic group is also cyclic - a cyclic subgroup.
$E$ is dense in $X$ if every point of $X$ is a limit point of $E,$ or a point of $E$ (or both.)
The derivative of a function $f$ at a point $a$ is defined as: $$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$ provided this limit exists.
The discrete metric is defined as:
$$ d(x, y) = \begin{cases} 0 & \text{if } x = y \\ 1 & \text{if } x \neq y \end{cases} $$
Two or more than two cycles are disjoint if no element appears in more than one cycle.
Let $R$ be a ring with unity. If every nonzero element of $R$ is a unit (has a multiplicative inverse), then $R$ is called a division ring.
A permutation of a finite set is even if it is the product of an even number of transpositions.
A commutative division ring is called a field.
A set $A$ is said to be finite if $A \sim \mathbb{N}_n$ for some $n.$
A group, is a set $G$ together with a binary operation $*$ such that:
Closure: The set is closed under the binary operation. For all $a, b \in G, a * b \in G$.
Associativity: The binary operation is associative on the set. For all $a, b, c \in G, (a * b) * c = a * (b * c)$.
Identity: The set contains an identity element, denoted $e$. For all $a \in G, a * e = a$.
Inverses: All elements in the set have inverse elements in the set, denoted using $a^{-1}$. For all $a \in G$ there exists $a^{-1} \in G$ such that $a * a^{-1} = e$.
This set/operation combination $G$ is commonly denoted as the pair $(G, *)$.
A half-open interval $(a,b]$ or $[a, b)$ is the set of all real numbers such that $a < x \leq b$ or $a \leq x < b,$ respectively.
A homomorphism is a map $\phi : G \to G'$ between groups (not necessarily a bijection), $\langle G, * \rangle$ and $\langle G', *' \rangle,$ that satisfies the homomorphism property:
$$ \phi(a * b) = \phi(a) *' \phi(b), ~ \forall a, b \in G. $$
The number of left cosets of a subgroup $H$ in a group $G$ is the index $(G:H)$ of $H$ in $G$.
A set $A$ is said to be infinite it is not finite.
A point $p$ is an interior point of $E$ if there is a neighborhood $N$ of $p$ such that $N \subset E.$
A interval $[a, b]$ is the set of all real numbers $x$ such that $a \leq x \leq b.$
If $p \in E$ and $p$ is not a limit point of $E,$ then $p is called an isolated point of E.
Given $\vec{a}, \vec{b} \in \mathbb{R}^k,$ if $\vec{a}_i < \vec{b}_i$ for all $i = 1, 2, \dots, k,$ then the set of all points $\vec{x}$ who satisfy $\vec{a}_i \leq \vec{x}_i \leq \vec{b}_i,$ $i = 1, 2, \dots, k,$ is called a k-cell. So, a 1-cell is an interval, a 2-cell is a rectangle, and so on.
The kernel of a homomorphism $\phi$ is the set of elements that $\phi$ sends to $e'$, and it is denoted by $\ker{(\phi)}$. It is a normal subgroup of $G$.
A point $p$ is a limit point of the set $E$ if every neighborhood of $p$ contains a point $q \neq p$ such that $q \in E.$
A set $X,$ whose elements we'll call points, together with a distance function $d: X \times X \to \mathbb{R}$ is called a metric space and the distance function $d$ is called a metric, if the following conditions, called the metric axioms, hold for $p, q, r \in X:$
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If $p \neq q, d(p,q) > 0.$ (distance is always positive between two distinct points.)
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$d(p,p) = 0.$ (distance is always zero between a point and itself.)
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$d(p,q) = d(q,p).$ (the distance from $p$ to $q$ is the same as the distance from $q$ to $p$.)
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$d(p,q) \leq d(p,r) + d(r,p)$ (triangle inequality.)
We can denote a metric space on set $X$ with metric $d$ as the tuple $(X, d).$
For some element $a$ in a ring with unity $R$ where $1 \neq 0$, if $a^{-1} \in R$ such that $aa^{-1} = a^{-1}a = 1$, $a^{-1}$ is said to be the multiplicative inverse of $a.$
A neighborhood, or r-neighborhood of $p$ is a set $N_r(p)$ consisting of all $q$ such that $d(p, q) < r$ for some $r > 0.$ This subset of $X$ is all the points within a circle of radius $r$ - the open ball of radius $r$ centered at $p.$
A subgroup $H$ of a group $G$ is normal if its left and right cosets coincide, that is, if $gH = Hg$, i.e. $gHg^{-1} = H$, for all $g \in G$.
A permutation of a finite set is odd if it is the product of an odd number of transpositions.
An open cover of a set $E$ in a metric space $X$ is a collection $\{G_\alpha\}$ of open subsets of $X$ such that $E \subset \bigcup_\alpha G_\alpha.$
Suppose $E \subset Y \subset X,$ and $X$ is a metric space. We say that $E$ is open relative to $Y$ if to each $p \in E$ there is associated an $r > 0$ such that $q \in E, q \in Y$ whenever $d(p, q) < r.$
$E$ is open if every point of $E$ is an interior point of $E.$
The order of a finite group is the number of its elements. The order of group $G$ is denoted as $\text{ord}{(G)}$ or $\|G\|$. The order of an element $a$ (also called period length or period) is the number of elements in the subgroup generated by $a$, and is denoted by $\text{ord}{(a)}$ or $\|a\|$.
$E$ is perfect if $E$ is closed and if every point of $E$ is a limit point of $E.$
A ring is a set together with two binary operations $+$ and $\cdot$, which we will call addition and multiplication, such that the following axioms are satisfied:
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Multiplication is associative.
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For all $a,b,c \in R$, the left distributive law and the right distribute law hold, i.e.
$$ a \cdot (b + c) = (a \cdot b) + (a \cdot c), \quad (a + b) \cdot c = (a \cdot c) + (b \cdot c). $$
A ring homomorphism $\phi : R \to R'$ must satisfy the following two properties:
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$\phi{(a+b)} = \phi{(a)} + \phi{(b)}.$
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$\phi{(ab)} = \phi{(a)}\phi{(b)}.$
A ring that has a multiplicative identity element is called a ring with unity.
A segment $(a, b)$ is the set of all real numbers $x$ such that $a < x < b.$
Two subsets $A$ and $B$ of a metric space $X$ are said to be separated if both $A \cap \overline{B}$ and $\overline{A} \cap B$ are empty, i.e., if no point of $A$ lies in the closure of $B$ and no point of $B$ lies in the closure of $A.$
A group $G$ is simple if it has no proper nontrivial normal subgroups, that is, if $|G| > 1$ and the only normal subgroups of $G$ are $\{e\}$ and $G$ itself.
A subgroup $H$ of a group $G$ is a subset of $G$ group together with the same operation as $G$ that still forms a group. The identity element of $G$ must also be the identity element of $H$.
A cycle of length 2 is a transposition.
A set $A$ is said to be uncountable if it is neither finite nor countable.
If $a$ has a multiplicative inverse in $R,$ $a$ is said to be a unit in $R$.
The element $1$ is also called unity.
A vector space over a field $F$ is a set $V$ together with two operations: - Vector addition: $V \times V \to V$ - Scalar multiplication: $F \times V \to V$
satisfying the following axioms: 1. $(V, +)$ is an abelian group 2. Scalar multiplication is associative: $a(bv) = (ab)v$ 3. Distributive laws hold 4. Identity: $1v = v$ for all $v \in V$