Engineering · Mathematics · Advanced Algebra Handbook
Cosets, Subgroup Index and Finite-Group Divisibility
Group theory studies algebraic symmetry through a set, a closed associative operation, an identity and inverses. The practical discipline is to move between elements, subgroups, maps, quotients and actions without losing the hypotheses that justify each step. This page consolidates the relevant material from the supplied algebra source into a stand-alone handbook chapter.
Executive summary
This chapter develops cosets, subgroup index and finite-group divisibility as part of a connected advanced-algebra learning sequence. The emphasis is on definitions, hypotheses, structural results and repeatable methods rather than historical narrative.
The source material is theorem-rich. Accordingly, the handbook presentation separates vocabulary from results and then adds a verification workflow so that each statement can be applied safely. Mathematical examples in the source are treated as examples, not as universal rules.
Problem-solving workflow
Identify the group, operation and identity, and decide whether additive or multiplicative notation is being used.
Determine the relevant subgroup and whether normality is required.
Use element order, cosets or a homomorphism to convert the question into a structural one.
When a quotient is involved, verify normality before forming cosets as group elements.
For an action, identify orbits, stabilisers, kernels and fixed points before counting.
Check the conclusion by tracing it back through the defining operation or map.
Core definitions
Definition
A subset H of a group G is a subgroup if (i) 1 ∈H; (ii) if x, y ∈H, then xy ∈H; (iii) if x ∈H, then x−1 ∈H. If H is a subgroup of G, we write H ≤G; if H is a proper subgroup of G, that is, H ̸= G, then we write H < G. subgroup-order theorem Observe that {1} and G are always subgroups of a group G, where {1} denotes the subset consisting of the single element 1. More interesting examples will be given soon. A subgroup H ̸= G is called a proper subgroup. Property (ii) shows that H is closed; that is, H has a binary operation. Associativity (xy)z = x(yz) holds for all x, y, z ∈G, and so this equation holds, in particular, for all x, y, z ∈H. Finally, (i) gives the identity, and (iii) gives inverses. It is easier to check that a subset H of a group G is a subgroup (and hence that it is a group in its own right) than to verify the group axioms for H: Associativity is inherited from the operation on G and hence it need not be verified again.
Definition
If G is a group and a ∈G, write ⟨a⟩= {an : n ∈Z} = {all powers of a}; ⟨a⟩is called the cyclic subgroup of G generated by a. A group G is called cyclic if there exists a ∈G with G = ⟨a⟩, in which case a is called a generator of G. Example 2.17(iv) shows, for every n ≥1, that the multiplicative group µn of all nth roots of unity is a cyclic group with the primitive nth root of unity ζ = e2πi/n as a generator. No doubt, the reader has seen the example of the integers modulo m in an earlier course. We merely recall the definition. . . , a −2m, a −m, a, a + m, a + 2m, . . . }. 7The alternating group first arose in studying polynomials. If f (x) = (x −u1)(x −u2) · · · (x −un), then the number D = i< j(ui −u j) can change sign when the roots are permuted: If α is a permutation of {u1, u2, . . . Thus, the sign of the product alternates as various permutations α are applied to its factors. subgroup-order theorem
Definition
The integers mod m, denoted8 by Im, is the family of all congruence classes mod m. Recall that [a] = [b] in Im if and only if a ≡b mod m. In particular, [a] = [0] in Im if and only if a ≡0 mod m; that is, [a] = [0] in Im if and only if m is a divisor of a. The definition of congruence mod m makes sense for all m ≥0, but the cases m = 0 and m = 1 are not very interesting: a ≡b mod 0 means 0 | (a −b), which says that a = b; a ≡b mod 1 means 1 | (a −b), which says that a and b are always congruent; that is, there is only one congruence class mod 1. Recall Proposition 1.19, which we now rewrite in the bracket notation.
Definition
If G is a finite group, then the number of elements in G, denoted by |G|, is called the order of G. The word order is used in two senses: the order of an element a ∈G and the order |G| of a group G. Proposition 2.34 shows that the order of a group element a is equal to | ⟨a⟩|.
Definition
If H is a subgroup of a group G and a ∈G, then the coset aH is the subset aH of G, where aH = {ah : h ∈H}. The cosets defined are often called left cosets; there are also right cosets of H, namely, subsets of the form Ha = {ha : h ∈H}. In general, left cosets and right cosets may be different, as we shall soon see. If we use the ∗notation for the operation in a group G, then we denote the coset aH by a ∗H, where a ∗H = {a ∗h : h ∈H}. In particular, if the operation is addition, then the coset is denoted by a + H = {a + h : h ∈H}. Of course, a = a1 ∈aH. Cosets are usually not subgroups. For example, if a /∈H, then 1 /∈aH (otherwise 1 = ah for some h ∈H, and this gives the contradiction a = h−1 ∈H).
Definition
The index of a subgroup H in G, denoted by [G : H], is the number of left10 cosets of H in G. The index [G : H] is the number t in the formula |G| = t|H| in the proof of subgroup-order theorem, so that |G| = [G : H]|H|; this formula shows that the index [G : H] is also a divisor of |G|; moreover, [G : H] = |G|/|H|.
Definition
If (G, ∗) and (H, ◦) are groups (we have displayed the operation in each), then a function f : G →H is a homomorphism11 if f (x ∗y) = f (x) ◦f (y) for all x, y ∈G. If f is also a bijection, then f is called an isomorphism. Two groups G and H are called isomorphic, denoted by G ∼= H, if there exists an isomorphism f : G → H between them. A multiplication table of a group G displays every product ab for a, b ∈G. G a1 a2 · · · a j · · · an a1 a1a1 a1a2 · · · a1a j · · · a1an a2 a2a1 a2a2 · · · a2a j · · · a2an ai aia1 aia2 · · · aia j · · · aian an ana1 ana2 · · · ana j · · · anan
Definition
Let a1, a2, . . . , an be a list with no repetitions of all the elements of a group G. A multiplication table for G is an n × n array whose i j entry is aia j. 11The word homomorphism comes from the Greek homo meaning “same” and morph meaning “shape” or “form.” Thus, a homomorphism carries a group to another group (its image) of similar form. The word isomorphism involves the Greek iso meaning “equal,” and isomorphic groups have identical form.
Principal results and structural facts
Key result
A subset H of a group G is a subgroup if and only if H is nonempty and, whenever x, y ∈H, then xy−1 ∈H.
Key result
A nonempty subset H of a finite group G is a subgroup if and only if H is closed; that is, if a, b ∈H, then ab ∈H. In particular, a nonempty subset of Sn is a subgroup if and only if it is closed.
Key result
Let m ≥2 be a fixed integer. (i) If a ∈Z, then [a] = [r] for some r with 0 ≤r < m. (ii) If 0 ≤r′ < r < m, then [r′] ̸= [r]. (iii) Im has exactly m elements, namely, [0], [1], . . . , [m −1]. For every m ≥2, Im is an (additive) cyclic group, where [a] + [b] = [a + b]; the identity is [0], the inverse of [a] is [−a], and a generator is [1]. Part (iii) shows that Im has order m. A cyclic group can have several different generators. For example, ⟨a⟩= a−1 .
Key result
(i) If G = ⟨a⟩is a cyclic group of order n, then ak is a generator of G if and only if (k, n) = 1. (ii) If G is a cyclic group of order n and gen(G) = {all generators of G}, then |gen(G)| = φ(n), where φ is the totient function.
Key result
Let G be a finite group and let a ∈G. Then the order of a is |⟨a⟩|, the number of elements in ⟨a⟩.
Key result
The intersection i∈I Hi of any family of subgroups of a group G is again a subgroup of G. In particular, if H and K are subgroups of G, then H ∩K is a subgroup of G.
Key result
If X is a subset of a group G, then there is a subgroup ⟨X⟩of G containing X that is smallest in the sense that ⟨X⟩≤H for every subgroup H of G that contains X.
Key result
, ⟨X⟩is a subgroup of G; of course, ⟨X⟩contains X because every H contains X. Finally, if H is any subgroup containing X, then H is one of the subgroups whose intersection is ⟨X⟩; that is, ⟨X⟩≤H. • Note that there is no restriction on the subset X in the last corollary; in particular, X = ∅ is allowed. Since the empty set is a subset of every set, we have ∅⊆H for every subgroup H of G. Thus, ⟨∅⟩is the intersection of all the subgroups of G; in particular, ⟨∅⟩≤{1}, and so ⟨∅⟩= {1}.
Key result
If X is a nonempty subset of a group G, then ⟨X⟩is the set of all the words on X.
Key result
Let H be a subgroup of a group G, and let a, b ∈G. (i) aH = bH if and only if b−1a ∈H. In particular, aH = H if and only if a ∈H. (ii) If aH ∩bH ̸= ∅, then aH = bH. (iii) |aH| = |H| for all a ∈G. subgroup-order theorem Remark. ◀
Key result
shows that the cosets partition G into pairwise disjoint subsets. It follows that |G| = |a1H| + |a2H| + · · · + |at H|. But |ai H| = |H| for all i, by Lemma 2.40(iii), so that |G| = t|H|, as desired. •
Key result
The set U(Im), defined by U(Im) = { [r] ∈Im : (r, m) = 1}, is a multiplicative group of order φ(m), where φ is the totient function. In particular, if p is a prime, then U(Ip) = I× p , the nonzero elements of Ip, is a multiplicative group of order p −1.
Source-grounded examples
Worked source example
(i) The four permutations V = { (1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3) } form a group, because V is a subgroup of S4 : (1) ∈V; α2 = (1) for each α ∈V, and so α−1 = α ∈V; the product of any two distinct permutations in V −{(1)} is the third one. The group V is called the four-group (V abbreviates the original German term Vierergruppe). Consider what verifying associativity a(bc) = (ab)c would involve: There are 4 choices for each of a, b, and c, and so there are 43 = 64 equations to be checked. Plainly, the best way to prove that V is a group is to show that it is a subgroup of S4. (ii) If R2 is the plane considered as an (additive) abelian group, then any line L through the origin is a subgroup. The reader may now verify that the axioms in the definition of subgroup do hold for L. ◀ We can shorten the list of items needed to verify that a subset is, in fact, a subgroup.
Worked source example
). A more interesting example is the strong resemblance between S3 and D6, the symmetries of an equilateral triangle. The notions of homomorphism and isomorphism allow us to compare different groups, as we shall see.
How to reason with these results
Most advanced-algebra problems become manageable when the representation is separated from the invariant structure. Begin with the definition, then decide whether the problem is asking for an elementwise calculation, a statement about a morphism, or a classification up to isomorphism. That choice determines the correct proof language.
When a theorem gives a structural conclusion, do not jump directly to the conclusion. Write the hypotheses next to the object you are studying and check them one by one. If a hypothesis fails, either strengthen the object, pass to a quotient or localisation where the theorem applies, or use a more elementary argument.
For computational work, record each transformation together with the equivalence relation it preserves. In algebra, row operations, similarity, quotienting, localisation and isomorphism preserve different kinds of information. A calculation is useful only when the preserved structure matches the question.
Common failure modes
| Failure mode | Control |
|---|---|
| Assuming a subgroup is normal because it is large or familiar. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Cancelling across a noncommutative product in the wrong order. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Confusing left and right cosets. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Assuming a homomorphism is injective or surjective without checking kernel or image. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
| Using an orbit-counting formula without confirming a genuine group action. | Return to the definition or theorem hypotheses and verify the missing condition before continuing. |
Verification checklist
- The ambient set, ring, field, group, module or category has been stated.
- Every operation and map used is well-defined in that setting.
- The hypotheses of each structural result have been checked before use.
- Representatives, coordinates or generators have not been confused with the underlying object.
- Existence and uniqueness have been separated where both matter.
- The final result has been checked against the original defining relation or universal property.
Quick questions
What should I identify first in a problem about cosets, subgroup index and finite-group divisibility?
Start with the ambient algebraic structure, its operation or maps, and the exact hypotheses. Most incorrect solutions begin by using a familiar rule that is not valid in the stated structure.
How should definitions be used in proofs?
Expand the definition at the point where it becomes useful. Definitions are not background prose; they are the conditions that determine what must be proved and which implications are available.
When is a structural theorem safer than direct calculation?
Use a structural theorem when its hypotheses are satisfied and the calculation would otherwise depend on arbitrary coordinates, representatives or generators. The theorem usually identifies an invariant that survives those choices.
How can a final answer be checked?
Substitute the result back into the defining relation, verify any required closure or map property, and check edge cases such as zero, the identity, the empty object or degenerate quotients where relevant.