The equal-or-disjoint theorem has a powerful consequence: all left cosets of a subgroup form a partition of the group, and all right cosets also form a partition. A partition is a decomposition into nonempty, pairwise disjoint subsets whose union is the whole set. In this lecture, cosets move from being individual subsets to being a complete organizational system for the group. This viewpoint is essential for index and Lagrange's theorem, where we count the number of cosets and multiply by the size of one coset. :::definition[Partition] Let $X$ be a nonempty set. A collection $\mathcal{P}$ of nonempty subsets of $X$ is called a $partition$ of $X$ if the following conditions hold: (i) every element of $X$ belongs to at least one member of $\mathcal{P}$; (ii) any two distinct members of $\mathcal{P}$ are disjoint. ::: :::theorem Let $(G,\circ)$ be a group and let $H$ be a subgroup of $G$. Then the set of all left cosets of $H$ in $G$ forms a partition of $G$. ::: :::proof Given that $(G,\circ)$ is a group and $H$ is a subgroup of $G$. To prove that the set of all left cosets of $H$ in $G$ forms a partition of $G$. [1] To prove that every element of $G$ belongs to at least one left coset of $H$. Let $a\in G$. Since $e\in H$, we get $$\begin{align*} a &=a\circ e\\ &\in aH. \end{align*}$$ Therefore every element of $G$ belongs to at least one left coset of $H$. [2] To prove that any two distinct left cosets are disjoint. Let $aH$ and $bH$ be two left cosets of $H$ in $G$. By the equal-or-disjoint theorem for left cosets, either $$\begin{equation*} aH=bH \end{equation*}$$ or $$\begin{equation*} aH\cap bH=\varnothing. \end{equation*}$$ Therefore two distinct left cosets must be disjoint. Hence, the set of all left cosets of $H$ in $G$ forms a partition of $G$. $\square$ ::: We observe how the left cosets of a subgroup divide $\mathbb{Z}_n$ into non-overlapping blocks. Choose a modulus and subgroup step, and the preview lists every distinct coset exactly once. Notice that every residue appears in one block and no residue appears in two different blocks. This is the partition idea that later supports index and Lagrange's theorem. :::scientific-preview[Coset Partition Explorer] ::: :::theorem Let $(G,\circ)$ be a group and let $H$ be a subgroup of $G$. Then the set of all right cosets of $H$ in $G$ forms a partition of $G$. ::: :::proof Given that $(G,\circ)$ is a group and $H$ is a subgroup of $G$. To prove that the set of all right cosets of $H$ in $G$ forms a partition of $G$. [1] To prove that every element of $G$ belongs to at least one right coset of $H$. Let $a\in G$. Since $e\in H$, we get $$\begin{align*} a &=e\circ a\\ &\in Ha. \end{align*}$$ Therefore every element of $G$ belongs to at least one right coset of $H$. [2] To prove that any two distinct right cosets are disjoint. Let $Ha$ and $Hb$ be two right cosets of $H$ in $G$. By the equal-or-disjoint theorem for right cosets, either $$\begin{equation*} Ha=Hb \end{equation*}$$ or $$\begin{equation*} Ha\cap Hb=\varnothing. \end{equation*}$$ Therefore two distinct right cosets must be disjoint. Hence, the set of all right cosets of $H$ in $G$ forms a partition of $G$. $\square$ ::: :::example[Partition of Integers Modulo Eight] Let $G=\mathbb{Z}_8$ under addition modulo $8$ and let $$\begin{equation*} H=\{0,4\}. \end{equation*}$$ The cosets of $H$ are $$\begin{align*} 0+H&=\{0,4\},\\ 1+H&=\{1,5\},\\ 2+H&=\{2,6\},\\ 3+H&=\{3,7\}. \end{align*}$$ These four subsets are nonempty, pairwise disjoint, and their union is $\mathbb{Z}_8$. Therefore they form a partition of $\mathbb{Z}_8$. ::: :::example[Representatives Are Not Unique] Let $G=\mathbb{Z}_8$ and $H=\{0,4\}$. Then $$\begin{align*} 1+H&=\{1,5\},\\ 5+H&=\{5,1\}. \end{align*}$$ Therefore $$\begin{equation*} 1+H=5+H. \end{equation*}$$ Hence, the same coset may have more than one representative. ::: :::solved-problem[Finding a Coset Partition] Let $G=\mathbb{Z}_{10}$ under addition modulo $10$ and let $H=\{0,5\}$. Find the partition of $G$ formed by the cosets of $H$. ::: :::solution[Solution] Let $G=\mathbb{Z}_{10}$ and $H=\{0,5\}$. The cosets are $$\begin{align*} 0+H&=\{0,5\},\\ 1+H&=\{1,6\},\\ 2+H&=\{2,7\},\\ 3+H&=\{3,8\},\\ 4+H&=\{4,9\}. \end{align*}$$ The remaining representatives repeat these cosets: $$\begin{align*} 5+H&=0+H,\\ 6+H&=1+H,\\ 7+H&=2+H,\\ 8+H&=3+H,\\ 9+H&=4+H. \end{align*}$$ Therefore the required partition is $$\begin{equation*} \{\{0,5\},\{1,6\},\{2,7\},\{3,8\},\{4,9\}\}. \end{equation*}$$ ::: We compute all distinct cosets of a subgroup of $\mathbb{Z}_n$. The calculator checks repeated representatives and keeps only one copy of each coset. The number of listed cosets is the number of blocks in the partition. Since every block has the same size, this is the counting pattern that later becomes Lagrange's theorem. :::calculator[Coset Partition Calculator] ::: :::exercise[Exercises] 1. Let $G=\mathbb{Z}_{12}$ and let $H=\{0,6\}$. Find the partition of $G$ formed by the cosets of $H$. 2. Let $G=\mathbb{Z}_{6}$ and let $H=\{0,3\}$. Decide whether $\{0,3\}$ and $\{1,4\}$ can overlap. 3. Let $(G,\circ)$ be a group and let $H$ be a subgroup of $G$. Explain why every element of $G$ belongs to some left coset of $H$. ::: :::answer[Answers] 1. The partition is $$\begin{equation*} \{\{0,6\},\{1,7\},\{2,8\},\{3,9\},\{4,10\},\{5,11\}\}. \end{equation*}$$ 2. They cannot overlap. They are distinct cosets of the same subgroup, so they are disjoint. 3. If $a\in G$, then $a=a\circ e$ and $e\in H$. Therefore $a\in aH$. ::: :::faq Q: What is the union of all left cosets of $H$? A: The union is the whole group $G$. Q: Can one element of $G$ belong to two different cosets? A: It may appear in two listed cosets only when those cosets are actually equal. Q: Why is the partition viewpoint useful? A: It lets us count a group by counting cosets and the number of elements in each coset. ::: :::call-to-action subtitle: Continue with algebraic tests for membership and equality of cosets. button: Next: Coset Membership and Equality Criteria | published/abstract-algebra/introduction-to-groups/cosets-of-subgroups/coset-membership-and-equality-criteria.md :::