The Set of Left (Right) Cosets of a Subgroup Partitions the Whole Group

The Set of Left (Right) Cosets of a Subgroup Partitions the Whole Group

Recall from the Left and Right Cosets of Subgroups page that if $(G, *)$ is a group, $(H, *)$ is a subgroup, and $g \in G$ then the left coset of $H$ with representative $g$ is the set:

(1)
\begin{align} \quad gH = \{ gh : h \in H \} \end{align}

The right coset of $H$ with representative $g$ is the set:

(2)
\begin{align} \quad Hg = \{ hg : h \in H \} \end{align}

We will now look at a nice theorem which tells us that the set of all left cosets of a subgroup $(H, *)$ actually partitions $(G, *)$. The proof below can be mirrored to analogously show that the set of all right cosets of a subgroup $(H, *)$ also partitions $(G, *)$.

Theorem 1: Let $(G, *)$ be a group and let $(H, *)$ a subgroup. Then all of the left cosets of $(H, *)$ partitions $(G, *)$.

Recall that a partition of a set $A$ is a collection of nonempty subsets of $A$ that are pairwise disjoint and whose union is all of $A$.

  • Proof: We first show that any two distinct left cosets of $H$ are disjoint. Let $g_1, g_2 \in G$, $g_1 \neq g_2$ and assume the left cosets $g_1H$ and $g_2H$ are distinct. Suppose that $g_1H \cap g_2H \neq \emptyset$. Then there exists an $x \in g_1H \cap g_2H$. So $x \in g_1H$ and $x \in g_2H$ and there exists $h_1, h_2 \in H$ such that:
(3)
\begin{align} \quad x = g_1h_1 \quad (*) \end{align}
(4)
\begin{align} \quad x = g_2h_2 \quad (**) \end{align}
  • So using $(*)$ and $(**)$ we see that $g_1h_1 = g_2h_2$. So $g_1 = g_2h_2h_1^{-1}$. But $h_2h_1^{-1} \in H$ since $(H, *)$ is a group and is hence closed under $*$. So $g_1 \in g_2H$. But this means that $g_1H = g_2H$, a contradiction since $g_1H$ and $g_2H$ distinct. So the assumption that $g_1H \cap g_2H \neq \emptyset$ was false. So:
(5)
\begin{align} \quad g_1H \cap g_2H = \emptyset \end{align}
  • Now if $i$ is the identity element for $(G, *)$ then $i \in H$ since $(H, *)$ is a subgroup and must contain the identity element. So $g = gi \in gH$ for all $g \in G$. So:
(6)
\begin{align} \quad G = \bigcup_{g \in G} gH \end{align}
  • Therefore the left cosets of $(H, *)$ partition $(G, *)$.
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