Left and Right Cosets of Subgroups

# Left and Right Cosets of Subgroups

 Definition: Let $(G, \cdot)$ be a group and let $(H, \cdot)$ be a subgroup. Let $g \in G$. Then the Left Coset of $H$ with Representative $g$ is the set $gH = \{ gh : h \in H \}$. The Right Coset of $H$ with Representative $g$ is the set $Hg = \{ hg : h \in H \}$.

When the operation symbol "$+$" is used instead of $\cdot$ we often denote the left and right cosets of $H$ with representation $g$ with the notation $g + H$ and $H + g$ respectively.

For example, consider the group $(\mathbb{Z}, +)$ and the subgroup $(3\mathbb{Z}, +)$. Consider the element $2 \in \mathbb{Z}$. Then the left coset of $3\mathbb{Z}$ with representative $2$ is:

(1)
\begin{align} \quad 2 + 3\mathbb{Z} = \{2 + h : h \in 3 \mathbb{Z} \} = \{ ..., -1, 2, 5, ... \} \end{align}

And the right coset of $3 \mathbb{Z}$ with representative $2$ is:

(2)
\begin{align} \quad 3\mathbb{Z} + 2 = \{ h + 2 : h \in 3 \mathbb{Z} \} = \{ ..., -1, 2, 5, ... \} \end{align}

In this particular example we see that $2 + 3\mathbb{Z} = 3\mathbb{Z} + 2$. But in general, is $gH = Hg$ for a given subgroup $(H, \cdot)$ of $(G, \cdot)$ and for $g \in G$? The answer is NO. There are many examples when left cosets are not equal to corresponding right cosets.

To illustrate this, consider the symmetric group $(S_3, \circ)$. Let $G = \{ \epsilon, (12) \}$. Then $(G, \circ)$ is a subgroup of $(S_3, \circ)$ since $G \subset S_3$, $G$ is closed under $\circ$, and $\epsilon^{-1} = \epsilon$, $(12)^{-1} = (12)$ (since $(12)$ is a transposition). Now consider the element $(13) \in S_3$. Then the left coset of $G$ with representative $(13)$ is:

(3)
\begin{align} \quad (13)G = \{ (13) \circ h : h \in G \} = \{ (13) \circ \epsilon, (13) \circ (12) \} = \{ (13), (123) \} \end{align}

And the right coset of $G$ with representative $(13)$ is:

(4)
\begin{align} \quad G(13) = \{ h \circ (13) : h \in G \} = \{ \epsilon \circ (13), (12) \circ (13) \} = \{ (13), (132) \} \end{align}

We note that $(123) \neq (132)$ and so $(13)G \neq G(13)$!

So, when exactly are the left and right cosets of a subgroup with representative $g$ equal? The following theorem gives us a simple criterion for a large class of groups.

 Proposition 1: Let $(G, \cdot)$ be a group and let $(H, \cdot)$ be a subgroup. If $(G, \cdot)$ is abelian then for all $g \in G$, $gH = Hg$.
• Proof: Let $g \in G$. If $G$ is abelian then for all $h \in G$ (and hence for all $h \in H$) we have that $g \cdot h = h \cdot g$. So:
(5)
\begin{align} \quad gH = \{ g \cdot h : h \in H \} = \{ h \cdot g : h \in H \} = Hg \quad \blacksquare \end{align}
 Proposition 2: Let $(G, \cdot)$ be a group, $(H, \cdot)$ a subgroup, and $g_1, g_2 \in G$. Then the following statements are equivalent: a) $g_1H = g_2H$. b) $Hg_1^{-1} = Hg_2^{-1}$. c) $g_1H \subseteq g_2H$. d) $g_1 \in g_2H$. e) $g_1^{-1}g_2 \in H$.