Subgroup, coset, quotient group, isomorphism theorems

Intuition [Beginner]

A subgroup is a smaller group living inside a bigger one. The even numbers sit inside all the integers: add two even numbers and you get an even number, zero is even, and the negative of an even number is even. The even numbers form a subgroup of the integers under addition.

A coset is a shifted copy of a subgroup. Start with the even numbers $\{0,2,4,\dots \}$ and shift everything by $1$: you get $\{1,3,5,\dots \}$. That shifted copy is a coset. The original subgroup and its cosets partition the whole group into disjoint blocks that tile everything with no overlap and no gaps.

A quotient group collapses each coset down to a single point. Instead of tracking individual numbers, you track which block they fall into. The integers modulo $n$ are the canonical example: collapse every block of $n$ consecutive integers to a single residue class.

Why does this concept exist? Subgroups, cosets, and quotients let you study a large group by breaking it into manageable pieces whose structure mirrors the original.

Visual [Beginner]

The integers partitioned into even and odd cosets. The even numbers (the subgroup) and the odd numbers (its single coset) form a two-block tiling of the number line.

Each block is a coset of the subgroup of even integers. The quotient group has exactly two elements: the even block and the odd block.

Worked example [Beginner]

Consider the integers under addition and the subgroup $3\mathbb{Z}=\{\dots ,-6,-3,0,3,6,\dots \}$ of multiples of $3$.

Step 1. The cosets are $0+3\mathbb{Z}=\{\dots ,-6,-3,0,3,6,\dots \}$, $1+3\mathbb{Z}=\{\dots ,-5,-2,1,4,7,\dots \}$, and $2+3\mathbb{Z}=\{\dots ,-4,-1,2,5,8,\dots \}$.

Step 2. Every integer falls into exactly one of these three blocks. Adding a number from block $0$ to a number from block $1$ always lands in block $1$.

Step 3. The quotient group $\mathbb{Z}/3\mathbb{Z}$ has three elements: the three blocks. Block addition works like clock arithmetic with three positions. Block $1$ plus Block $2$ equals Block $0$.

What this tells us: collapsing a subgroup produces a new group whose elements are cosets and whose operation inherits from the original.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $G$ be a group with identity $e$.

Definition (Subgroup). A subset $H\subseteq G$ is a subgroup if $H$ is itself a group under the operation inherited from $G$. Equivalently, $H$ is non-empty and $a{b}^{-1}\in H$ for all $a,b\in H$ (the one-step subgroup test). Notation: $H\le G$ ^{[Dummit-Foote §3]}.

Definition (Left coset). For $H\le G$ and $g\in G$, the left coset of $H$ containing $g$ is $$ gH = {gh : h \in H}. $$ Right cosets are defined analogously as $Hg=\{hg:h\in H\}$.

Definition (Index). The index $[G:H]$ is the number of distinct left cosets of $H$ in $G$. This may be infinite.

Definition (Normal subgroup). A subgroup $N\le G$ is normal if $gN{g}^{-1}=N$ for every $g\in G$. Equivalently, $gN=Ng$ for all $g\in G$: left and right cosets coincide. Notation: $N⊴G$.

Definition (Quotient group). When $N⊴G$, the set of cosets $G/N=\{gN:g\in G\}$ forms a group under the operation $(gN)(hN)=ghN$. The identity is $eN=N$, and the inverse of $gN$ is $g^{-1}N$ ^{[Lang §I.3-4]}.

Counterexamples to common slips [Intermediate+]

• A subgroup is not the same as a subset. The subset $\{0,1\}$ of $\mathbb{Z}$ is not a subgroup: $1+1=2$ is not in $\{0,1\}$.
• Left cosets and right cosets need not coincide. In the symmetric group $S_3$, take $H=\{e,(12)\}$. Then $(13)H=\{(13),(132)\}$ but $H(13)=\{(13),(123)\}$.
• The quotient set $G/H$ is a group only when $H$ is normal. The coset multiplication $(gH)(kH)=gkH$ is well-defined precisely when left and right cosets coincide.

Key theorem with proof [Intermediate+]

Theorem (Lagrange). Let $G$ be a finite group and $H\le G$. Then $|H|$ divides $|G|$, and $[G:H]=|G|/|H|$.

Proof. Define the relation $\sim$ on $G$ by $g\sim k$ if $g^{-1}k\in H$.

Reflexivity: $g^{-1}g=e\in H$, so $g\sim g$.

Symmetry: if $g^{-1}k\in H$, then $(g^{-1}k{)}^{-1}=k^{-1}g\in H$, so $k\sim g$.

Transitivity: if $g^{-1}k\in H$ and $k^{-1}m\in H$, then $g^{-1}m=(g^{-1}k)(k^{-1}m)\in H$, so $g\sim m$.

Thus $\sim$ is an equivalence relation. The equivalence class of $g$ is $gH$: for $g^{-1}k\in H$ write $k=g(g^{-1}k)$ with $g^{-1}k\in H$. The cosets partition $G$ into disjoint sets.

The map $H\to gH$ given by $h\mapsto gh$ is a bijection (its inverse is $gh\mapsto h$). Every coset has exactly $|H|$ elements. If there are $[G:H]$ cosets and they partition $G$, then $|G|=[G:H]\cdot |H|$. $\square$

Bridge. Lagrange's theorem builds toward 03.03.02 (group action), where the orbit-stabiliser theorem reproduces the same counting argument in a broader setting. This result appears again in 01.02.01 (group), where the order of an element divides the order of the group as an immediate corollary. The foundational reason is that cosets tile the group with equal-size blocks, and this is exactly the mechanism behind every divisibility result in finite group theory. The central insight is that the subgroup's order measures the internal symmetry of the group, and the bridge is between that measurement and the arithmetic of the group's order.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (First Isomorphism Theorem). Let $\phi :G\to H$ be a group homomorphism. Then $\ker \phi ⊴G$ and $G/\ker \phi \cong \mathrm{im}\phi$.

The isomorphism is $\bar{\phi }:gK\mapsto \phi (g)$ where $K=\ker \phi$. Well-definedness follows from $\phi (gk)=\phi (g)\phi (k)=\phi (g)$ for $k\in K$ ^{[Dummit-Foote §3]}.

Theorem 2 (Second Isomorphism Theorem). Let $N⊴G$ and $H\le G$. Then $H\cap N⊴H$ and $H/(H\cap N)\cong HN/N$.

This is the algebraic analogue of the parallelogram law for vector subspaces: $|H|+|N|=|HN|+|H\cap N|$ when all groups are finite.

Theorem 3 (Third Isomorphism Theorem). Let $N⊴G$ and $K⊴G$ with $N\subseteq K$. Then $K/N⊴G/N$ and $(G/N)/(K/N)\cong G/K$.

This is the cancellation law for quotients: quotienting by $N$ and then by $K/N$ is the same as quotienting directly by $K$ ^{[Lang §I.3-4]}.

Theorem 4 (Correspondence Theorem). Let $N⊴G$. There is a bijection between subgroups of $G$ containing $N$ and subgroups of $G/N$, given by $H\mapsto H/N$. Under this bijection, normal subgroups correspond to normal subgroups.

This identifies the lattice of intermediate subgroups between $N$ and $G$ with the lattice of subgroups of the quotient $G/N$.

Theorem 5 (Jordan-Holder). Let $G$ be a finite group. Any two composition series $$ {e} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_r = G $$ have the same length $r$, and the composition factors $G_{i+1}/G_i$ are the same up to permutation.

A group is simple if its only normal subgroups are $\{e\}$ and itself. Simple groups are the indecomposable building blocks of finite group theory. The Jordan-Holder theorem guarantees that the multiset of simple composition factors is an invariant of $G$ ^{[Rotman §3]}.

Theorem 6 (Cauchy). If $G$ is a finite group and a prime $p$ divides $|G|$, then $G$ contains an element of order $p$.

Cauchy's theorem is a partial converse to Lagrange: while not every divisor of $|G|$ is the order of a subgroup (as the alternating group $A_4$ shows for the divisor $6$), every prime divisor is.

Synthesis. Lagrange's theorem is the foundational reason that the arithmetic of group orders constrains subgroup structure, and this is exactly the starting point for every counting argument in the theory. The three isomorphism theorems generalise the familiar rank-nullity picture from linear algebra: the First Isomorphism Theorem identifies the quotient by a kernel with the image, the Second identifies the quotient of $H$ by its overlap with $N$, and the Third identifies the quotient of quotients. Putting these together with the Correspondence Theorem, the bridge is between the subgroup lattice of $G/N$ and the intermediate subgroups of $G$ containing $N$. The central insight is that normal subgroups are kernels of homomorphisms, and the pattern generalises across algebra: rings modulo ideals, modules modulo submodules, and topological spaces modulo identifications all follow the same isomorphism-theorem template.

Full proof set [Master]

Proposition 1 (First Isomorphism Theorem). Let $\phi :G\to H$ be a homomorphism. Then $G/\ker \phi \cong \mathrm{im}\phi$.

Proof. Set $K=\ker \phi$. Define $\bar{\phi }:G/K\to \mathrm{im}\phi$ by $\bar{\phi }(gK)=\phi (g)$.

Well-defined: if $gK=g^{\prime }K$ then $g^{-1}{g}^{\prime }\in K$, so $\phi (g^{-1}{g}^{\prime })=e$, giving $\phi (g)=\phi (g^{\prime })$.

Homomorphism: $\bar{\phi }(gK\cdot hK)=\bar{\phi }(ghK)=\phi (gh)=\phi (g)\phi (h)=\bar{\phi }(gK)\bar{\phi }(hK)$.

Injective: $\bar{\phi }(gK)=e$ implies $\phi (g)=e$, so $g\in K$, hence $gK=K$ is the identity coset.

Surjective: for $h\in \mathrm{im}\phi$ choose $g$ with $\phi (g)=h$; then $\bar{\phi }(gK)=h$.

Therefore $\bar{\phi }$ is an isomorphism. $\square$

Proposition 2 (Third Isomorphism Theorem). Let $N⊴G$ and $K⊴G$ with $N\subseteq K$. Then $(G/N)/(K/N)\cong G/K$.

Proof. Define $\psi :G/N\to G/K$ by $\psi (gN)=gK$. This is well-defined because $N\subseteq K$ ensures $gN=g^{\prime }N$ implies $g^{-1}{g}^{\prime }\in N\subseteq K$, so $gK=g^{\prime }K$.

Homomorphism: $\psi (gN\cdot hN)=\psi (ghN)=ghK=(gK)(hK)=\psi (gN)\psi (hN)$.

Surjective: for any $gK\in G/K$, $\psi (gN)=gK$.

Kernel: $\ker \psi =\{gN:gK=K\}=\{gN:g\in K\}=K/N$.

By the First Isomorphism Theorem applied to $\psi$, $(G/N)/(K/N)\cong G/K$. $\square$

Proposition 3 (Correspondence Theorem). Let $N⊴G$. The maps $H\mapsto H/N$ (from subgroups of $G$ containing $N$ to subgroups of $G/N$) and $\bar{H}\mapsto {\pi }^{-1}(\bar{H})$ (from subgroups of $G/N$ to subgroups of $G$ containing $N$) are mutually inverse bijections preserving normality, index, and inclusion.

Proof. Let $\pi :G\to G/N$ be the canonical projection. For a subgroup $H$ of $G$ containing $N$, the image $\pi (H)=H/N$ is a subgroup of $G/N$ (it is closed under the quotient operation because $H$ is closed). For a subgroup $\bar{H}$ of $G/N$, the preimage ${\pi }^{-1}(\bar{H})$ is a subgroup of $G$ containing $N$ (it contains $N$ because $\pi (N)=\{N\}$, the identity of $G/N$).

These maps are mutually inverse: ${\pi }^{-1}(\pi (H))=H$ because $H$ contains $N$ and the fibers of $\pi$ over $H/N$ are exactly the cosets $hN$ with $h\in H$. Conversely, $\pi ({\pi }^{-1}(\bar{H}))=\bar{H}$ because $\pi$ is surjective onto $G/N$.

If $H⊴G$ with $N\subseteq H$, then $H/N⊴G/N$: for any $gN\in G/N$ and $hN\in H/N$, $(gN)(hN)(gN{)}^{-1}=gh{g}^{-1}N\in H/N$ since $gh{g}^{-1}\in H$. Conversely, if $\bar{H}⊴G/N$, then ${\pi }^{-1}(\bar{H})⊴G$ because $\pi$ is a homomorphism and normality pulls back through surjective homomorphisms. $\square$

Connections [Master]

• Group 01.02.01. This unit builds directly on the definition of a group, its identity element, and its operation. The subgroup, coset, and quotient constructions here are the structural machinery that the earlier group unit introduces at the axiomatic level. The foundational reason that normal subgroups correspond to kernels of homomorphisms appears again in the group unit's advanced results as a preview.

• Subspace, basis, dimension 01.01.04. The analogy between subgroups and subspaces runs deep: both are closed subsets of an ambient algebraic structure, both admit cosets or translates, and both yield quotient objects (quotient groups and quotient vector spaces). The First Isomorphism Theorem for groups parallels the rank-nullity theorem for linear maps, and this is exactly the pattern that linear algebra units develop for vector spaces.

• Group action 03.03.02. Lagrange's theorem generalises to the orbit-stabiliser theorem when a group acts on a set: the size of an orbit equals the index of the stabiliser. The coset decomposition in Lagrange's proof is the same partition that orbits induce. The quotient group $G/\mathrm{Stab}(x)$ maps injectively into the orbit of $x$, and this connection identifies coset spaces with homogeneous spaces in the Lie-group setting.

• Dirichlet $L$-functions 21.03.02. The multiplicative group $(\mathbb{Z}/m\mathbb{Z}{)}^{\times }$ — central to the definition of a Dirichlet character — is the unit group of the residue ring obtained from the quotient construction of the present unit applied to $\mathbb{Z}$ modulo $m\mathbb{Z}$. The Chinese remainder theorem then identifies it as a product ${\prod }_{p^k\Vert m}(\mathbb{Z}/p^k\mathbb{Z}{)}^{\times }$ of cyclic and near-cyclic factors, and Dirichlet characters factor through this structural decomposition; the number of Dirichlet characters modulo $m$ is exactly $\phi (m)$, the order of this unit group.

Historical & philosophical context [Master]

Galois around 1830 introduced the concept of a normal subgroup (which he called a "proper decomposition") in his work on the solvability of polynomial equations by radicals ^{[Galois 1830]}. The quotient construction arose implicitly: Galois observed that when a group of permutations admits a "proper decomposition," the substitution groups form a new group. Jordan systematised these ideas in his 1870 Traite des substitutions, establishing the isomorphism theorems and the correspondence between normal subgroups and quotient groups in the language of permutation groups ^{[Jordan 1870]}.

Holder proved the Jordan-Holder theorem on composition series in 1889, establishing that the simple factors of a finite group form a well-defined invariant ^{[Rotman §3]}. Noether reoriented group theory around homomorphisms and quotients in the 1920s, making the isomorphism theorems central to the abstract-algebra curriculum. The modern three-theorem formulation (First, Second, Third Isomorphism Theorems) follows her Göttingen lectures and their influence on van der Waerden's Moderne Algebra (1930–31).

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