Notation Key

Notation	Meaning
$Z$ , $Z$	the integers
$Q$ , $Q$	the rational numbers
$R$ , $R$	the real numbers
$C$ , $C$	the complex numbers
$Z_{n}$ , $Z / n Z$ , $Z / n Z$	the integers modulo $n$
$\ker (T)$ , $nul (T)$	kernel (or null space) of the linear transformation $T$
$im (T)$ , $range (T)$	image (or range) of the linear transformation $T$
$M_{n} (R)$ , $Mat (n; R)$	$n \times n$ matrices with entries in the ring $R$
$V^{⊥}$	orthogonal complement of the subspace $V$
$H \leq G$	$H$ is a subgroup of $G$
$[G : H]$	index of the subgroup $H$ in $G$
$g H$	left coset of the subgroup $H$ represented by $g$
$G / H$	set of left cosets of the subgroup $H$ in $G$
$K ⊴ G$	$K$ is a normal subgroup of $G$
$Z (G)$	center of the group $G$
$S_{n}$	symmetric group on $n$ letters
$S_{X}$	permutation group of the set $X$
$A_{n}$	alternating group on $n$ letters
$D_{2 n}$ , $D_{n}$	dihedral group of order $2 n$ (the symmetries of the regular $n$ -gon)
$K_{4}$ , $V$ , $Z_{2} \times Z_{2}$	Klein 4-group
$Aut (G)$	automorphism group of the group $G$
$Inn (G)$	inner automorphism group of the group $G$
$R^{\times}$	multiplicative group of units of a ring
$ch (R)$ , $char (R)$	characteristic of the ring $R$
$R [x_{1}, \dots, x_{n}]$	polynomial ring in variables $x_{1}, \dots, x_{n}$ with coefficients in the ring $R$
$Z [i]$ , $Z [i]$	Gaussian integers
$⟨ A ⟩$ , $(A)$ , $R A$	(left) ideal generated by the subset $A$ of the ring $R$

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