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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 5.30
\(n!\) \(n\) factorial Example 6.34
\(\binom{n}{k}\) binomial coefficient \(n!/(k!(n-k)!)\) Example 6.34
\(a \mid b\) \(a\) divides \(b\) Paragraph
\(\gcd(a, b)\) greatest common divisor of \(a\) and \(b\) Paragraph
\(\mathbb Z_n\) the integers modulo \(n\) Paragraph
\(U(n)\) group of units in \(\mathbb Z_n\) Example 2.11
\(\mathbb M_n(\mathbb R)\) the \(n \times n\) matrices with entries in \(\mathbb R\) Example 2.14
\(\det A\) the determinant of \(A\) Example 2.14
\(GL_n(\mathbb R)\) the general linear group Example 2.14
\(Q_8\) the group of quaternions Example 2.15
\(\mathbb C^*\) the multiplicative group of complex numbers Example 2.16
\(|G|\) the order of a group Paragraph
\(\mathbb R^*\) the multiplicative group of real numbers Example 3.9
\(\mathbb Q^*\) the multiplicative group of rational numbers Example 3.9
\(SL_n(\mathbb R)\) the special linear group Example 3.11
\(Z(G)\) the center of a group Exercise 3.3.15
\(\langle a \rangle\) cyclic group generated by \(a\) Theorem 4.3
\(|a|\) the order of an element \(a\) Paragraph
\(\cis \theta\) \(\cos \theta + i \sin \theta\) Paragraph
\(\mathbb T\) the circle group Paragraph
\(S_n\) the symmetric group on \(n\) letters Paragraph
\((a_1, a_2, \ldots, a_k )\) cycle of length \(k\) Paragraph
\(A_n\) the alternating group on \(n\) letters Paragraph
\(D_n\) the dihedral group Paragraph
\(G \cong H\) \(G\) is isomorphic to a group \(H\) Paragraph
\(\aut(G)\) automorphism group of a group \(G\) Exercise 6.3.37
\(i_g\) \(i_g(x) = gxg^{-1}\) Exercise 6.3.41
\(\inn(G)\) inner automorphism group of a group \(G\) Exercise 6.3.41
\(\rho_g\) right regular representation Exercise 6.3.44
\([G:H]\) index of a subgroup \(H\) in a group \(G\) Paragraph
\(\mathcal L_H\) the set of left cosets of a subgroup \(H\) in a group \(G\) Theorem 7.8
\(\mathcal R_H\) the set of right cosets of a subgroup \(H\) in a group \(G\) Theorem 7.8
\(a \notdivide b\) \(a\) does not divide \(b\) Theorem 7.19
\(G/N\) factor group of \(G\) mod \(N\) Paragraph
\(G'\) commutator subgroup of \(G\) Exercise 9.3.14
\(\ker \phi\) kernel of \(\phi\) Paragraph