Examples of groups

Symbol	Operation	Identity	Elements	Inverse	Abelian
$ℤ$	addition	0	$𝑘$	$- 𝑘$	yes
$ℚ +$	multiplication	1	$𝑚 / 𝑛$	$𝑛 / 𝑚$	yes
$ℤ 𝑛$	addition mod $𝑛$	0	$𝑘$	$𝑛 - 𝑘$	yes
$ℝ ⋆$	multiplication	1	$𝑥$	$𝑥 - 1$	yes
$ℂ ⋆$	multiplication	1	$𝑎 + 𝑏 𝑖$	$(𝑎 + 𝑏 𝑖) - 1$	yes
$𝐺 𝐿 (2, 𝐹)$	matrix multiplication	$𝐼$	matrices	by Gaußian elimination	no
$ℤ \times 𝑛$	multiplication mod $𝑛$	1	$𝑘$ coprime with $𝑛$	by Euclid’s algorithm	yes
$ℝ 𝑛$	componentwise addition	$⃗ 𝟎$	$⃗ 𝐯$	$- ⃗ 𝐯$	yes
$𝑆 𝐿 (2, 𝐹)$	matrix multiplication	$𝐼$	matrices of determinant 1	inverse matrix	yes
$𝐷 𝑛$	composition	$𝑅 0$	$𝑅 𝛼, 𝐿$	$𝑅 2 𝜋 - 𝛼, 𝐿$	no
$𝑆 𝑂 (3)$	orthogonal rotation	$𝐼$	rotation (matrices)	inverse rotation (matrix)

Particular groups