Cosets

Let H be a subgroup of a group G. If a ∈ G, then the set 
                                       Ha = {ha : h ∈ H}
is called a right coset of H in G determined by a and the set
                                       aH = {ah : h ∈ H}
is called a left coset of H in G determined by a.
If the operation in addition, then the right coset becomes
                                       H + a = {h + a : h ∈ H}
and the left coset becomes
                                      a + H = {a + h : h ∈ H}.

If e is the identity element of group G, then He and eH are right and left cosets of H in G.
Also    He = {he : h ∈ H} = {h : h ∈ H} = H.
            eH = {eh : h ∈ H} = {h : h ∈ H} = H.
∴   If H is a subgroup of a group G, then H itself is a right coset as well as left coset of H in G determined by e.

Remark: When G is an abelian group then there is no distinction between a left coset and a right coset.
i.e., left coset = right coset.
i.e., aH = Ha.

General Properties: The identity is in precisely one left or right coset, namely H itself. Thus H is both a left and right coset of itself.
A coset representative is a representative in the equivalence class sense. A set of representatives of all the cosets is called a transversal. There are other types of equivalence relations in a group, such as conjugacy, that form different classes, which do not have the properties discussed here.

Example: Find the right cosets of the subgroup {1, -1} of the group {1, -1, i, -i} under multiplication.
Sol. G = {1, -1, i, -i} is a group under multiplication.
         H = {1, -1} is a subgroup of G.
The right cosets of H in G are H1, H(-1), Hi, H(-i)
         H · 1  = { 1(1), -1(1) } = { 1, -1 } = H.
         H(-1) = { 1(-1), -1(-1) } = { -1, 1 } = H.
         Hi      = { 1(i), -1(i) }
         H(-i)  = { 1(-i), -1(-i) } = {-i , i}.

Example: Let <S₃ , 0> be the symmetric group on 
{1, 2, 3} and H = {I, (12)} is subgroup of <S₃ , 0>. Find all the left cosets of H in S₃ and also find all the right cosets of H in S₃.
Sol. Let S₃ = {i, (12), (13), (23), (123), (132)} be a symmetric group on {1, 2, 3} and H = {I, (12)}.
        The Left cosets of H in S₃ are
           I · H = H
         (12)H = {(12)I, (12)(12)} = {(12), I} = H
         (13)H = {(13)I, (13)(12)} = {(13), (132)}
         (23)H = {(23)I, (23)(12)} = {(23), (123)}
        (123)H = {(123)I, (123)(12)} = {(123), (23)}
        (132)H = {(132)I, (132)(12)} = {(132), (13)}
∴  The distinct left cosets of H in S₃ are H, (13)H and (23)H.
The right cosets of H in S₃ are
          H · I = H
        H(12)  = {I(12), (12)(12)} = {(12), I} = H
        H(13)  = {I(13), (12)(13)} = {(13), (123)}
        H(23)  = {I(23), (12)(23)} = {(23), (132)}
        H(123) = {I(123), (12)(123)} = {(123), (13)}
        H(132) = {I(132), (12)(132)} = {(132), (23)}
The distinct right cosets of H in S₃ are H, H(13) and H(23).

Example: Let G be the group of integers under addition. Let H be the subgroup of G having even integers. Find the right cosets of H in G.
Sol. G = Group of all integers under addition.
                H = {o, ±2, ±4, ...} is subgroup of G having even integers.
         H + 0 = {h + 0 : h ∈ H} = {h : h ∈ H} = H
         H + 1 = {h + 1 : h ∈ H} = {±1, ±3, ...}
Similarly,
        H + 2 = {o, ±2, ±4, ...}
        H + 3 = {±1, ±3, ...}  and so on.
∴ The only distinct right cosets of H in G are H and H + 1.