If
H ≤ G and a and b belong to G, then
(1)
(2)
aH = H if and only if 
(3)
aH = bH
or aH ∩ bH
= Ø
(4)
aH = bH
if and only if 
(5)
|aH| = |bH|
(6)
aH = Ha if and only if H = aHa-1
(7)
aH ≤ G if and only if
If
H ≤ G and a and b belong to G, then
(1)
(2)
aH = H if and only if
(3)
aH = bH
or aH ∩ bH
= Ø
(4)
aH = bH
if and only if
(5)
|aH| = |bH|
(6)
aH = Ha if and only if H = aHa-1
(7)
aH ≤ G if and only if