Skip to main content
\(\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{{\not{\mid}}} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)


The following table defines the notation used in this book. Page numbers or references refer to the first appearance of each symbol.

Symbol Description Location
\(a \in A\) \(a\) is in the set \(A\) Paragraph
\({\mathbb N}\) the natural numbers Paragraph
\({\mathbb Z}\) the integers Paragraph
\({\mathbb Q}\) the rational numbers Paragraph
\({\mathbb R}\) the real numbers Paragraph
\({\mathbb C}\) the complex numbers Paragraph
\(A \subset B\) \(A\) is a subset of \(B\) Paragraph
\(\emptyset\) the empty set Paragraph
\(A \cup B\) the union of sets \(A\) and \(B\) Paragraph
\(A \cap B\) the intersection of sets \(A\) and \(B\) Paragraph
\(A'\) complement of the set \(A\) Paragraph
\(A \setminus B\) difference between sets \(A\) and \(B\) Paragraph
\(A \times B\) Cartesian product of sets \(A\) and \(B\) Paragraph
\(A^n\) \(A \times \cdots \times A\) (\(n\) times) Paragraph
\(id\) identity mapping Paragraph
\(f^{-1}\) inverse of the function \(f\) Paragraph
\(a \equiv b \pmod{n}\) \(a\) is congruent to \(b\) modulo \(n\) Example 1.2.30
\(n!\) \(n\) factorial Example 2.1.4
\(\binom{n}{k}\) binomial coefficient \(n!/(k!(n-k)!)\) Example 2.1.4
\(a \mid b\) \(a\) divides \(b\) Paragraph
\(\gcd(a, b)\) greatest common divisor of \(a\) and \(b\) Paragraph
\(\mathcal P(X)\) power set of \(X\) Exercise 2.3.12
\(\lcm(m,n)\) the least common multiple of \(m\) and \(n\) Exercise 2.3.23
\(\mathbb Z_n\) the integers modulo \(n\) Paragraph
\(U(n)\) group of units in \(\mathbb Z_n\) Example 3.2.4
\(\mathbb M_n(\mathbb R)\) the \(n \times n\) matrices with entries in \(\mathbb R\) Example 3.2.7
\(\det A\) the determinant of \(A\) Example 3.2.7
\(GL_n(\mathbb R)\) the general linear group Example 3.2.7
\(Q_8\) the group of quaternions Example 3.2.8
\(\mathbb C^*\) the multiplicative group of complex numbers Example 3.2.9
\(|G|\) the order of a group Paragraph
\(\mathbb R^*\) the multiplicative group of real numbers Example 3.3.1
\(\mathbb Q^*\) the multiplicative group of rational numbers Example 3.3.1
\(SL_n(\mathbb R)\) the special linear group Example 3.3.3
\(Z(G)\) the center of a group Exercise 3.4.48
\(\langle a \rangle\) cyclic group generated by \(a\) Theorem 4.1.3
\(|a|\) the order of an element \(a\) Paragraph
\(\cis \theta\) \(\cos \theta + i \sin \theta\) Paragraph
\(\mathbb T\) the circle group Paragraph