Cosets, Normal Subgroups, Quotient Groups and Homomorphisms
Cosets, Normal Subgroups, Quotient Groups and Homomorphisms: core definitions, structural results and verification methods in abstract algebra.
How the topic fits together
Cosets, Normal Subgroups, and Homomorphisms
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Definition
Let H be a subgroup of the group G. If g ∈G, the right coset of H generated by g is Hg = {hg : h ∈H}; similarly, the left coset of H generated by g is gH = {gh : h ∈H}. It follows from the definitions (Problem 1) that if a, b ∈G, then Ha = Hb if and only if ab−1 ∈H, and aH = bH if and only if a−1b ∈H. Thus if define a and b to be equivalent iffab−1 ∈H, one has an equivalence relation (Problem 2), and the equivalence class of a is (Problem 3) {b : ab−1 ∈H} = Ha.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Lagrange’s Theorem
Lagrange’s Theorem If H is a subgroup of G, then |G| = |H|[G : H]. In particular, if G is finite then |H| divides |G|, and |G| |H| = [G : H].
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Corollary
Let G be a finite group. (i) If a ∈G then |a| divides |G|; in particular, a|G| = 1. Thus |G| is a multiple of the order of each of its elements, so if define the exponent of G to be the least common multiple of {|a| : a ∈G}, then |G| is a multiple of the exponent. (ii) If G has prime order, then G is cyclic.
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Euler’s Theorem
Euler’s Theorem If a and n are relatively prime positive integers, with n ≥2, then aϕ(n) ≡1 mod n. A special case is Fermat’s Little Theorem: if p is a prime and a is a positive integer not divisible by p, then ap−1 ≡1 mod p.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
The Index is Multiplicative
The Index is Multiplicative If K ≤H ≤G then [G : K] = [G : H][H : K].
Proof / verification strategy: Use subgroup generation, coset/orbit counting and divisibility of orders. For cyclic groups, reduce the claim to arithmetic on exponents modulo the group order.
Proposition
If H ≤G and K ≤G, then HK ≤G if and only if HK = KH. In this case, HK is the subgroup generated by H ∪K.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Lemma
If H ≤G, then (aH)(bH) = abH for all a, b ∈G iffcHc−1 = H for all c ∈G. (Equivalently, cH = Hc for all c ∈G.)
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Definition
Let H be a subgroup of G. If any of the following equivalent conditions holds, call H is normal subgroup of G, or that H is normal in G: 1. cHc−1 ⊆H for all c ∈G (equivalently, c−1Hc ⊆H for all c ∈G) 2. cHc−1 = H for all c ∈G (equivalently, c−1Hc = H for all c ∈G) 3. cH = Hc for all c ∈G 4. Every left coset of H in G is also a right coset 5. Every right coset of H in G is also a left coset One has established the equivalence of 1,2 and 3 above, and 3 immediately implies 4.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Definition of the Quotient Group If H is normal in G, we may define a group
of the Quotient Group If H is normal in G, one may define a group multiplication on cosets, as follows. If aH and bH are (left) cosets, let (aH)(bH) = abH; by (1.3.7), (aH)(bH) is simply the set product. If a1 is another member of aH and b1 another member of bH, then a1H = aH and b1H = bH (Problem 5). Therefore the set product of a1H and b1H is also abH.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Example
Let GL(n, R) be the set of all nonsingular n by n matrices with real coefficients, and let SL(n, r) be the subgroup formed by matrices whose determinant is 1 (GL stands for “general linear” and SL for “special linear”). Then SL(n, R) ◁GL(n, R), because if A is a nonsingular n by n matrix and B is n by n with determinant 1, then det(ABA−1) = (det A)(det B)(det A−1) = det B = 1.
Proof / verification strategy: Treat this as an illustration of the surrounding definitions. Recompute the stated relations directly and keep the numerical or structural choices local to the example.
Definition
If f : G →H, where G and H are groups, then f is called a homomorphism if for all a, b in G, one has f(ab) = f(a)f(b). This idea will look familiar if G and H are abelian, in which case we write, using additive notation, f(a + b) = f(a) + f(b); thus a linear transformation on a vector space is, in particular, a homomorphism on the underlying abelian group. If f is a homomorphism from G to H, it must map the identity of G to the identity of H, since f(a) = f(a1G) = f(a)f(1G); multiply by f(a)−1 to get 1H = f(1G). Furthermore, the inverse of f(a) is f(a−1), because 1 = f(aa−1) = f(a)f(a−1), so that [f(a)]−1 = f(a−1).
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
The Connection Between Homomorphisms and Normal Subgroups
The Connection Between Homomorphisms and Normal Subgroups If f : G →H is a homomorphism, define the kernel of f as kerf = {a ∈G : f(a) = 1}; then kerf is a normal subgroup of G. For if a ∈G and b ∈kerf, we must show that aba−1 belongs to kerf. But f(aba−1) = f(a)f(b)f(a−1) = f(a)(1)f(a)−1 = 1. Conversely, every normal subgroup is the kernel of a homomorphism.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Proposition
A homomorphism f is injective if and only if its kernel K is trivial, that is, consists only of the identity.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Proposition
Let f : G →H be a homomorphism. (i) If K is a subgroup of G, then f(K) is a subgroup of H. If f is an epimorphism and K is normal, then f(K) is also normal. (ii) If K is a subgroup of H, then f −1(K) is a subgroup of G.
Proof / verification strategy: Work directly from closure, inverses, subgroup tests and the defining action or presentation. Use a small cyclic or permutation model to verify the direction of the argument.
Quick-reference relationships
Problem-solving workflow
Identify the ambient group
State the operation, identity, inverses and whether commutativity is available.
Locate the relevant subgroup structure
Check generated subgroups, normality, cosets, stabilisers or direct factors before applying a theorem.
Use the correct counting or mapping tool
Choose coset counting, orbit counting, a homomorphism, a quotient or a group action as appropriate.
Check hypotheses explicitly
Finite-order assumptions, normality, prime-power divisibility and coprimality conditions are not interchangeable.
Translate the result back to structure
Interpret the result as a statement about subgroup size, quotient structure, action, decomposition or presentation.
Verify with a small model
Use a cyclic, symmetric, dihedral or modular example to test the logic without treating the example as the theorem.
Worked-solution emphasis from the supplied source
The source exercises reinforce definition checking, construction of examples and direct use of the formal results in this article.
Common mistakes and boundary conditions
- Treating left and right cosets as identical without normality.
- Assuming the converse of a subgroup-order divisibility result.
- Confusing the order of a group with the order of one of its elements.
- Using quotient multiplication before checking that the subgroup is normal.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 1.3 | Cosets, Normal Subgroups, and Homomorphisms | 15–19 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.