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Appendix A Notation

Symbol Description Location
\(x\in A\) set membership Definition 0.1.1
\(A\subseteq B\) set inclusion Definition 0.1.3
\(A\cup B\) set union Definition 0.1.8
\(A\cap B\) set intersection Definition 0.1.8
\(A-B\) set difference Definition 0.1.8
\(A^c\) set complement Definition 0.1.8
\(\{\ \}, \emptyset\) the empty set Definition 0.1.9
\(\mathbb{R}\) real numbers Definition 0.1.9
\(\mathbb{Z}\) integers Definition 0.1.9
\(\Z_{+}\) positive integers Definition 0.1.9
\(\Z_{\geq a}\) the set of all integers greater than or equal to \(a\) Definition 0.1.9
\(\mathbb{Q}\) rational numbers Definition 0.1.9
\(\mathcal{P}(A)\) power set of \(A\) Definition 0.1.10
\(f\colon A\rightarrow B\) a function from \(A\) to \(B\) Definition 0.1.12
\(f(A)\) image of the set \(A\) under \(f\) Definition 0.1.17
\(\operatorname{im} f\) image of a function \(f\) Definition 0.1.17
\(f^{-1}(A)\) preimage of \(A\) under \(f\) Definition 0.1.18
\(f\circ g\) the composition of \(f\) and \(g\) Definition 0.1.21
\(f=(f(i))_{i\in I}\) tuple indexed by \(I\) Definition 0.1.24
\(\prod_{i\in I}A_i\) Cartesian product of the sets \(A_i\) Definition 0.1.33
\(A^n\) \(n\)-fold Cartesian product of \(A\) Definition 0.1.33
\(A^\infty\) Cartesian product \(A^{\Z_+}\) Definition 0.1.33
\(\abs{A}\) cardinality of set \(A\) Definition 0.1.34
\(G\times H\) product of groups \(G\) and \(H\) Definition 1.1.5
\(\prod_{i\in I} G_i\) product of family of groups \((G_i)_{i\in I}\) Definition 1.1.5
\(a\mid b\) \(a\) divides \(b\) Definition 1.2.1
\(a\equiv b\pmod n\) \(a\) is congruent to \(b\) modulo \(n\) Definition 1.2.2
\(\overline{a}, [a]_n\) the congruence class of \(a\) modulo \(n\) Definition 1.2.2
\(M_n(R)\) \(n\times n\) matrices with coefficients in \(R\) Definition 1.3.1
\(\GL_n(R)\) the group of invertible \(n\times n\) matrices with coefficients in \(R\) Definition 1.3.1
\(\Isom(\R^n)\) isometries of \(\R^n\) Definition 1.4.1
\(D_n\) the dihedral group of cardinality \(2n\) Specimen 8
\(Q_8\) the quaternion group Specimen 9
\(S_A\) permutations of the set \(A\) Definition 1.5.1
\(H\leq G\) \(H\) is a subgroup of \(G\) Definition 1.8.1
\(C_G(A)\) centralizer of \(A\) in \(G\) Definition 1.8.12
\(N_G(A)\) normalizer of \(A\) in \(G\) Definition 1.8.12
\(\angvec{A}\) subgroup generated by set \(A\) Definition 1.10.2
\(G\colon H\) index of \(H\) in \(G\) Definition 1.11.9
\(\sgn\) the sign map Definition 1.16.3
\(G_X\) group action stabilizer of \(X\) Definition 1.17.2
\(v_p(n)\) \(p\)-adic valuation of \(n\) Definition 1.20.1
\(\Aff(\Z/n\Z)\) group of invertible affine transformations of \(\Z/n\Z\) Specimen 15
\(G_1\rtimes G_2\) semidirect product of \(G_1\) and \(G_2\) Specimen 16
\(f^\phi\) change of coefficients of \(f\) under \(\phi\) Definition 2.5.9
\(f\boldmod I\) reduction of \(f\) modulo \(I\) Definition 2.5.9
\((A)\) the ideal generated by \(A\) Definition 2.5.16
\(\ch R\) characteristic of the ring \(R\) Definition 2.5.20
\(S^{-1}R\) localization of ring \(R\) by \(S\) Definition 2.10.1
\(\Frac R\) the fraction field of \(R\) Definition 2.10.7
\(R[x_1,x_2,\dots, x_n]\) multivariate polynomial ring over \(R\) Specimen 32
\(\boldx^{\boldi}\) multivariate monomial notation Definition 2.15.2
\(\N\) the natural numbers \(\Z_{\geq 0}\) Definition 2.15.2
\(\Hom_R(M,N)\) set of \(R\)-module homomorphisms from \(M\) to \(N\) Definition 2.17.7
\(\End_R(M)\) set of \(R\)-module endomorphisms of \(M\) Definition 2.17.7
\(\Ann M\) annihilator of module \(M\) Definition 2.19.1
\(\Tor M\) torsion elements of module \(M\) Definition 2.19.1
\(M(\alpha)\) the \(\alpha\)-torsion of a module \(M\text{.}\) Definition 2.19.6
\(M(\pi^\infty)\) the \(\pi\)-primary component of a module \(M\text{.}\) Definition 2.19.6