Representations, Characters and Finite-Group Applications
Noncommutative algebra requires explicit control of multiplication order, one-sided ideals and module conventions. Representation theory then encodes group or algebra structure through linear actions and characters. This page consolidates the relevant material from the supplied algebra source into a stand-alone handbook chapter.
Learning pathNoncommutative Algebra and Representations
LevelAdvanced
FormatHandbook guide
Read time12 min
Executive summary
This chapter develops representations, characters and finite-group applications 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
Fix left-versus-right conventions before beginning.
Keep multiplication order unchanged unless commutativity has been proved.
Use one-sided ideals, radicals and chain conditions appropriate to the setting.
Decompose modules into simple pieces only when the semisimplicity hypotheses apply.
For representations, identify the field, dimension, kernel and invariant subspaces.
Use characters as class functions only after verifying the representation-theoretic assumptions.
Core definitions
Definition
A representation of a group G is a homomorphism σ : G →GL(V ), where V is a vector space over C. The degree of σ is dim(V ). For the remainder of this section, we restrict ourselves to finite groups and representations having finite degree. If a representation σ : G →GL(V ) has degree n and one chooses a basis of V , then each σ(g) can be regarded as an n × n nonsingular matrix with entries in C. Representations can be translated into the language of modules. In Proposition 8.37, we proved that every representation σ : G →GL(V ) equips V with the structure of a left CG-module (and conversely): If g ∈G, then σ(g): V →V , and we define scalar multiplication gv, for g ∈G and v ∈V , by gv = σ(g)(v).
Definition
A representation σ of a group G is irrreducible if the corresponding CGmodule is simple; a representation σ is completely reducible if it is a direct sum of irreducible representations; that is, the corresponding CG-module is semisimple.
Definition
A function ϕ : G →C is a class function if it is constant on conjugacy classes; that is, if h = xgx−1, then ϕ(h) = ϕ(g). Characters Every character χσ afforded by a representation σ is a class function: If h = xgx−1, then σ(h) = σ(xgx−1) = σ(x)σ(g)σ(x)−1, and so tr(σ(h)) = tr(σ(g)); that is, χσ(h) = χσ(g). Not every class function is a character. For example, if χ is a character, then −χ is a class function; it is not a character because −χ(1) is negative, and so it cannot be a degree.
Definition
If α, β ∈cf(G), define (α, β) = |G| g∈G α(g)β(g), where c denotes the complex conjugate of a complex number c. Note that (α, α) is real, by (ii), and the inner product is definite; that is, (α, α) > 0 if α ̸= 0.
Definition
If χτ is the character afforded by a representation τ : G →GL(V ), then ker χτ = ker τ.
Definition
If H is a subgroup of a group G, then every representation σ : G →GL(V ) gives, by restriction, a representation σ|H : H →GL(V ). (In terms of modules, every left CG-module V can be viewed as a left CH-module.) We call σ|H the restriction of σ, and we denote it by σ⇃H. The character of H afforded by σ⇃H is denoted by χσ⇃H. The next result displays an interesting relation between characters on a group and characters on a subgroup. (Formally, it resembles the adjoint isomorphism.)
Definition
A doubly transitive G-set X is sharply doubly transitive if only the identity of G fixes two elements of X; that is, Gx,y = {1} for all distinct pairs x, y ∈X.
Definition
Let M be an R-module. If m ∈M, then its order ideal (or annihilator) is ann(m) = {r ∈R : rm = 0}. We say that m has finite order (or is a torsion1 element) if ann(m) ̸= {0}; otherwise, m has infinite order. When a commutative ring R is regarded as a module over itself, its identity 1 has infinite order, for ann(1) = {0}.
Principal results and structural facts
Key result
Every group G of order pmqn, where p and q are primes, is a solvable group. Notice that finite-group solvability theorem cannot be improved to groups having orders with only three distinct prime factors, for A5 is a simple group of order 60 = 22 · 3 · 5. Using representations, we will prove the following theorem. Theorem. If G is a nonabelian finite simple group, then {1} is the only conjugacy class whose size is a prime power.
Key result
(i) Every irreducible representation of a finite group G is equivalent to one of the representations λi given in Proposition 8.119(i). (ii) Every irreducible representation of a finite abelian group is linear. (iii) If σ : G →GL(V ) is a representation of a finite group G, then σ(g) is similar to a diagonal matrix for each g ∈G.
Key result
(i) Every character χσ is an N-linear combination of the irreducible characters χi = χλi afforded by λi : G →GL(Li): there are integers mi ≥0 with χσ = i miχi. (ii) Equivalent representations have the same character. (iii) The only irreducible characters of G are χ1, . . . , χr.
Key result
With respect to the inner product just defined, the irreducible characters χ1, . . . , χr form an orthonormal basis; that is, (χi, χ j) = δi j.
Key result
says that the weighted inner product of distinct rows in the character table is 0, while the weighted inner product of any row with itself is 1. Characters
Key result
(i) |G| = r i=1 n2 i (ii) r i=1 niχi(gk) = 0 if k > 1 (iii) r k=1 hkχi(gk) = 0 if i > 1 (iv) r k=1 hk|χi(gk)|2 = |G|
Key result
Let θ = χτ be the character of a finite group G afforded by a representation τ : G →GL(V ). (i) For each g ∈G, we have |θ(g)| ≤θ(1). (ii) ker θ = {g ∈G : θ(g) = θ(1)}. (iii) If θ = j m jχ j, where m j are positive integers, then ker θ = " j ker χ j. (iv) If N is a normal subgroup of G, then there are irreducible characters χi1, . . ., χis with N = s j=1 ker χi j .
Key result
Let H be a subgroup of a finite group G and let χ be a character on H. (i) χ↿G(1) = [G : H]χ(1). (ii) If H ✁G, then χ↿G(g) = 0 for all g /∈H.
Key result
Let H be a subgroup of a group G, let χ be a class function on G, and let θ be a class function on H. Then (θ↿G, χ)G = (θ, χ⇃H)H, where ( , )G denotes the inner product on cf(G) and ( , )H denotes the inner product on cf(H). Characters
Key result
If σ : G →GL(V ) is an irreducible representation and if a linear transformation ϕ : V →V satisfies ϕσ(g) = σ(g)ϕ for all g ∈G, then ϕ is a scalar transformation: there exists α ∈C with ϕ = α1V . Theorems of finite-group solvability and of fixed-point-action
Key result
If G is a nonabelian finite simple group, then {1} is the only conjugacy class whose size is a prime power. Therefore, finite-group solvability theorem is true: every group of order pmqn, where p and q are primes, is solvable.
Key result
If X is a doubly transitive G-set, then |G| = n(n −1)|Gx,y|, where n = |X| and Gx,y = {g ∈G : gx = x and gy = y}. Moreover, if X is a faithful G-set, then |Gx,y| is a divisor of (n −2)!.
Key result
A finite group G is a fixed-point-action groups if and only if it contains a proper nontrivial subgroup H such that H ∩gHg−1 = {1} for all g /∈H.
Key result
Let G be a fixed-point-action groups with fixed-point-action complement H and fixed-point-action kernels N. Then N is a normal subgroup of G, N ∩H = {1}, and N H = G. Remark. A group G having a subgroup Q and a normal subgroup K such that K ∩Q = {1} and K Q = G is called a semidirect product. ◀
Source-grounded examples
Worked source example
We now show that permutation representations, that is, G-sets,9 give a special kind of representation. A G-set X corresponds to a homomorphism π : G →SX, where SX is the symmetric group of all permutations of X. If V is the complex vector space having X as a basis, then we may regard SX ≤GL(V ) in the following way. Each permutation 9Recall that if a group G acts on a set X, then X is called a G-set. Characters π(g) of X, where g ∈G, is now a permutation of a basis of V and, hence, it determines a nonsingular linear transformation on V . With respect to the basis X, the matrix of π(g) is a permutation matrix: It arises by permuting the columns of the identity matrix I by π(g); thus, it has exactly one entry equal to 1 in each row and column while all its other entries are 0. ◀ One of the most important representations is the regular representation; in terms of modules, the regular representation is the group algebra CG regarded as a left module over itself.
Worked source example
(i) The symmetric group S3 is a fixed-point-action groups: X = {1, 2, 3} is a faithful transitive S3-set; no α ∈(S3)# fixes two elements; each transposition (i j) fixes one element. The cyclic subgroups ⟨(i j)⟩are fixed-point-action complements (so fixed-point-action complements need not be unique). A permutation β ∈S3 has no fixed points if and only if β is a Theorems of finite-group solvability and of fixed-point-action 3-cycle. We are going to prove that, in every fixed-point-action groups, 1 together with all those elements having no fixed points comprise a normal subgroup. (ii) The example of S3 in part (i) can be generalized. Let X be a G-set, with at least three elements, which is a sharply doubly transitive G-set. Then X is transitive, Gx,y = {1}, and Gx ̸= {1} (for if x, y, z ∈X are distinct, there exists g ∈G with x = gx and z = gy). Therefore, every sharply doubly transitive group G is a fixed-point-action groups. ◀
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
Silently commuting factors.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Confusing left modules with right modules.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Assuming every module decomposes into simples.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Using character identities outside the required field or finiteness conditions.
Return to the definition or theorem hypotheses and verify the missing condition before continuing.
Treating a matrix representation as faithful without checking its kernel.
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 representations, characters and finite-group applications?
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.
Connections within the handbook
Source basis: supplied advanced algebra reference. Source-identifying authorship, publisher information, acknowledgements and biographical material are intentionally omitted. Mathematical terminology and results are retained in handbook form.
