Quotient group

$Congruence$

a

and

b

are left congruence:

a \sim_{L G H} b : a^{- 1} b \in H

a

and

b

are right congruence:

a \sim_{RG H} b : b a^{- 1} \in H

L G H, RG H

are equivalence relations

G

is a left residue class by

H :

[a]_{L}^{G H} = {x \in G ∣ a \sim_{L G H} b}

G

is a right residue class by

H :

[a]_{R}^{G H} = {x \in G ∣ a \sim_{RG H} b}

[a]_{L}^{G H} = a H

[a]_{R}^{G H} = H a

Expected $

Left complete system of representatives of

G

about

H

:

{g_{1}, g_{2}, ... g_{n}} (g_{i} \in [s_{i}]_{L}^{G H})

Right complete system of representatives of

G

about

H

:

{g_{1}, g_{2}, ... g_{m}} (g_{i} \in [t_{i}]_{R}^{G H})

{a_{λ}}_{λ \in Λ}

is a left(right) complete system of representatives of

G

about

H

\Leftrightarrow

{a_{λ}^{- 1}}_{λ \in Λ}

is a left(right) complete system of representatives of

G

about

H

{a_{λ} b_{μ}}_{λ \in Λ, μ \in M}

is a left(right) complete system of representatives of

G

about

H

[a]_{L}^{GN} = a N = N a = [a]_{R}^{GN}

{a_{λ}}_{λ \in Λ}

is left complete system of reprensentatives of

G

about

N

\Leftrightarrow {a_{λ}}_{λ \in Λ}

is right complete system of reprensentatives of

G

abount

N