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.
Learning pathGroup Theory
LevelAdvanced
FormatHandbook guide
Read time11 min
Executive summary
This chapter develops group homomorphisms and isomorphism principles 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
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. A multiplication table of a group G of order n depends on how we list the elements of G, and so G has n! different multiplication tables. (Thus, the task of determining whether a multiplication table of a group G is the same as some multiplication table of another group H is a daunting one: It involves about n! comparisons, each of which involves comparing n2 entries.) If a1, a2, . . . , an is a list of all the elements of G with no repetitions, and if f : G →H is a bijection, then f (a1), f (a2), . . . , f (an) is a list of all the elements of H with no repetitions, and this latter list determines a multiplication table for H. That f is an isomorphism says that if we superimpose the given multiplication table for G (determined by a1, a2, . . . , an) upon the multiplication table for H [determined by f (a1), f (a2), . . . , f (an)], then the tables match: If aia j is the i j entry in the given multiplication table of G, then f (ai) f (a j) = f (aia j) is the i j entry of the multiplication table of H. In this sense, isomorphic groups have the same multiplication table. Thus, isomorphic groups are essentially the same, differing only in the notation for the elements and the operations.
Definition
If f : G →H is a homomorphism, define kernel12 f = {x ∈G : f (x) = 1} and image f = {h ∈H : h = f (x) for some x ∈G}. We usually abbreviate kernel f to ker f and image f to im f . 12Kernel comes from the German word meaning “grain” or “seed” (corn comes from the same word). Its usage here indicates an important ingredient of a homomorphism.
Definition
A subgroup K of a group G is called a normal subgroup if k ∈K and g ∈G imply gkg−1 ∈K. If K is a normal subgroup of G, we write K ✁G. The proposition thus says that the kernel of a homomorphism is always a normal subgroup. If G is an abelian group, then every subgroup K is normal, for if k ∈K and g ∈G, then gkg−1 = kgg−1 = k ∈K. The converse of this last statement is false: In Example 2.63, we shall show that there is a nonabelian group (the quaternions), each of whose subgroups is normal. The cyclic subgroup H = ⟨(1 2)⟩of S3, consisting of the two elements (1) and (1 2), is not a normal subgroup of S3: If α = (1 2 3), then α−1 = (3 2 1), and α(1 2)α−1 = (1 2 3)(1 2)(3 2 1) = (2 3) /∈H [by Theorem 2.9, α(1 2)α−1 = (α1 α2) = (2 3)]. On the other hand, the cyclic subgroup K = ⟨(1 2 3)⟩of S3 is a normal subgroup, as the reader should verify. It follows from Examples 2.55(i) and 2.55(ii) that An is a normal subgroup of Sn and SL(n, R) is a normal subgroup of GL(n, R) (however, it is also easy to prove these facts directly).
Definition
If G is a group and a ∈G, then a conjugate of a is any element in G of the form gag−1, where g ∈G. It is clear that a subgroup K ≤G is a normal subgroup if and only if K contains all the conjugates of its elements: If k ∈K, then gkg−1 ∈K for all g ∈G. Homomorphisms
Definition
If G is a group and g ∈G, define conjugation γg : G →G by γg(a) = gag−1 for all a ∈G.
Definition
The group of quaternions14 is the group Q of order 8 consisting of the following matrices in GL(2, C): Q = { I, A, A2, A3, B, B A, B A2, B A3 }, where I is the identity matrix, A = −1 , and B = i i . The element A ∈Q has order 4, so that ⟨A⟩is a subgroup of order 4 and hence of index 2; the other coset is B ⟨A⟩= {B, B A, B A2, B A3 }. Thus, every element in Q has an expression of the form Bi A j, where i = 0, 1 and j = 0, 1, 2, 3.
Principal results and structural facts
Key result
Let f : G →H be a homomorphism of groups. (i) f (1) = 1 (ii) f (x−1) = f (x)−1 (iii) f (xn) = f (x)n for all n ∈Z
Key result
Let f : G →H be a homomorphism. (i) ker f is a subgroup of G and im f is a subgroup of H. (ii) If x ∈ker f and if a ∈G, then axa−1 ∈ker f . (iii) f is an injection if and only if ker f = {1}.
Key result
(i) If G is a group and g ∈G, then conjugation γg : G →G is an isomorphism. (ii) Conjugate elements have the same order.
Key result
(i) If H is a subgroup of index 2 in a group G, then g2 ∈H for every g ∈G. (ii) If H is a subgroup of index 2 in a group G, then H is a normal subgroup of G.
Source-grounded examples
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.
Worked source example
shows that Q is a nonabelian group that is like abelian groups in that every subgroup is normal. This is essentially the only such example. A nonabelian finite group is called hamiltonian if every subgroup is normal; every hamiltonian group has the form Q × A, where A is an abelian group with no elements of order 4 (direct products will be introduced in the next section). subgroup-order theorem states that the order of a subgroup of a finite group G must be a divisor of |G|. This suggests the question, given a divisor d of |G|, whether G must contain a subgroup of order d. The next result shows that there need not be such a subgroup.
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 group homomorphisms and isomorphism principles?
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.
