# AppendixBNotation

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 |