1 Definition and basic concepts
A subgroup is a subset of a group that forms a group under the same operation. The notion isolates parts of a group that retain the full algebraic structure, making it possible to study a large group through its smaller, more manageable pieces. Subgroups appear throughout group theory because they encode internal symmetry and support many standard constructions.
1.1 Groups and subsets
A group is a set equipped with an operation that is associative, has an identity element, and gives each element an inverse. A subgroup begins as a subset of such a set, but not every subset is compatible with the group operation. The key question is whether the subset remains closed under the operation and inversion and contains the identity.
1.2 Subgroup definition
A subset H of a group G is a subgroup if H itself is a group with the operation inherited from G. This means that the operation on H is not newly defined; it is the same multiplication or composition already used in G. The subgroup condition is therefore a test of whether the ambient structure restricts cleanly to the subset.
1.2.1 Closure under the group operation
Closure under the operation means that combining any two elements of the subset produces another element of the same subset. For a multiplicative group, if a and b lie in H, then ab must also lie in H. Without this property, the subset cannot preserve the algebraic structure of the larger group.
1.2.2 Closure under inverses
Every element of a subgroup must have its inverse inside the subgroup. If a is in H, then a^-1 must also be in H. This requirement ensures that equations solvable in the larger group remain solvable within the smaller one.
1.2.3 Identity element criterion
A subgroup must contain the identity element of the ambient group. This follows from the group axioms, since the identity is necessary for the subset to function as a group in its own right. In many practical tests, showing closure under the operation and inverses is enough because the identity then follows automatically.
1.3 Subgroup tests
Rather than checking all axioms directly, algebraists often use shorter criteria to verify subgroup status. These tests reduce the amount of work needed and are especially useful when dealing with concrete examples such as matrix groups or transformation groups.
1.3.1 Two-step test
A common criterion states that a nonempty subset H of a group G is a subgroup if it is closed under products and inverses. If a and b are in H, then ab and a^-1 must also belong to H. This is one of the most widely used methods for establishing that a subset is a subgroup.
1.3.2 One-step test
A more compact version says that a nonempty subset H is a subgroup if for any a and b in H, the element ab^-1 is also in H. This single condition combines closure under multiplication and inverses in one statement. It is particularly efficient in theoretical arguments.
1.3.3 Finite subset criterion
When the ambient group is finite, a nonempty subset closed under the group operation is automatically a subgroup. In a finite setting, closure under multiplication forces the presence of inverses and the identity. This criterion is useful in computational and classification problems.
2 Examples of subgroups
Subgroups arise in many familiar settings, from arithmetic groups to matrix groups and symmetry groups. These examples illustrate how the general definition applies across different algebraic contexts.
2.1 Trivial and whole group subgroups
Every group has two immediate subgroups: the trivial subgroup containing only the identity and the whole group itself. These are sometimes called the improper subgroups because they do not give a proper internal decomposition. Even so, they play an important role as boundary cases in many theorems.
2.2 Cyclic subgroups
Given an element of a group, the set of all its powers forms a subgroup called a cyclic subgroup. Cyclic subgroups are among the simplest and most important examples, since they are generated by a single element.
2.2.1 Generators and powers
If g is an element of a group, then the cyclic subgroup generated by g consists of all integer powers of g. In additive notation, this corresponds to all integer multiples of an element. The element g is called a generator of that subgroup.
2.2.2 Finite and infinite cyclic groups
A cyclic subgroup may be finite or infinite depending on whether some positive power of the generator returns the identity. If this happens, the subgroup has finite order; otherwise, it is infinite. Finite cyclic groups are structurally simple and are determined by their order.
2.3 Matrix subgroups
Matrix groups provide many natural examples of subgroups because matrix multiplication is associative and invertibility is well understood. Subgroups in this setting often reflect geometric or linear-algebraic constraints.
2.3.1 Special linear and diagonal subgroups
The special linear group consists of matrices with determinant 1, and it forms a subgroup of the general linear group. Diagonal invertible matrices also form a subgroup, since products and inverses of diagonal matrices remain diagonal. These examples show how algebraic conditions can define meaningful subgroups.
2.3.2 Upper triangular subgroups
The set of invertible upper triangular matrices is a subgroup of the general linear group. Multiplying two upper triangular matrices preserves upper triangular form, and the inverse of an invertible upper triangular matrix is again upper triangular. Such subgroups are common in linear algebra and representation theory.
2.4 Subgroups in permutation groups
Permutation groups contain many subgroups formed by restricting allowed rearrangements. For example, permutations that fix a chosen element make a subgroup, as do collections of permutations preserving a partition of a set. These examples are central in the study of symmetry and combinatorial structure.
3 Basic properties
Subgroups have several structural properties that make them easy to combine and compare. These properties support the lattice-like organization of the subgroup structure of a group.
3.1 Intersection of subgroups
The intersection of any collection of subgroups of a group is again a subgroup. This follows because the common elements inherit closure properties from each subgroup containing them. Intersections are therefore a basic way to build smaller subgroups from larger ones.
3.2 Generated subgroup
Given a subset of a group, one can form the smallest subgroup containing it. This construction produces the subgroup generated by that subset and is fundamental in describing how groups are built from selected elements.
3.2.1 Subgroup generated by a set
The subgroup generated by a set S is the set of all finite products of elements of S and their inverses. It is the smallest subgroup that contains S. In practice, this subgroup captures all consequences of starting with the chosen elements.
3.2.2 Minimality property
The generated subgroup is minimal among subgroups containing the original set. Any subgroup that contains S must also contain the subgroup generated by S. This universal property makes generation a central organizing principle in group theory.
3.3 Subgroup lattice
The family of all subgroups of a given group can be organized by inclusion. This arrangement is called the subgroup lattice and records how subgroups sit inside one another.
3.3.1 Inclusion relations
If one subgroup is contained in another, the smaller one is called a subgroup of the larger. Chains of inclusions reveal layers of structure and often guide classification. The inclusion order allows subgroups to be compared systematically.
3.3.2 Maximal and minimal subgroups
A maximal subgroup is a proper subgroup not contained in any larger proper subgroup. A minimal nontrivial subgroup has no smaller nontrivial subgroup inside it. These extremal cases are important in structural analysis and finite group theory.
4 Cosets and index
Cosets describe how a subgroup partitions a group into translated copies. They are essential tools for counting and for constructing quotient groups.
4.1 Left cosets
Given a subgroup H of a group G and an element g in G, the left coset of H determined by g is the set gH. It consists of all products gh with h in H. Left cosets are either identical or disjoint, and together they partition the group.
4.2 Right cosets
Right cosets are defined similarly by Hg, the set of products hg with h in H. In general, left and right cosets need not coincide, though they do in many common cases such as abelian groups. Right cosets provide a complementary way to view the same partitioning idea.
4.3 Index of a subgroup
The index of a subgroup H in G is the number of distinct cosets of H in G. It measures how many translated copies of H are needed to cover the whole group. The index can be finite or infinite.
4.3.1 Finite index
A subgroup has finite index when only finitely many cosets occur. Finite index often signals that the subgroup is large relative to the group. Such subgroups play a major role in counting arguments and in the study of finite quotients.
4.3.2 Counting cosets
In a finite group, the size of the group equals the product of the size of a subgroup and its index. This relation is a cornerstone of finite group theory and is often used to deduce divisibility properties. It also connects subgroup structure with enumeration.
5 Normal subgroups
Normal subgroups are subgroups that behave well under conjugation and are exactly the subgroups that can be used to form quotient groups. They are among the most important special types of subgroups.
5.1 Definition of normality
A subgroup N of G is normal if gNg^-1 = N for every element g of G. This condition means that conjugating elements of N by elements of the whole group leaves N unchanged. Normality ensures compatibility with the ambient symmetry.
5.2 Characterizations of normal subgroups
Normal subgroups can be recognized in several equivalent ways. These characterizations make it easier to identify normality in different contexts and connect it to homomorphisms and symmetry.
5.2.1 Conjugation invariance
A subgroup is normal precisely when it is stable under conjugation by every group element. This expresses the idea that the subgroup is internally consistent with the action of the whole group. Conjugation invariance is often the most direct criterion in concrete examples.
5.2.2 Kernels of homomorphisms
Every kernel of a group homomorphism is a normal subgroup. Conversely, every normal subgroup arises as the kernel of some homomorphism. This correspondence links normality with maps between groups and gives normal subgroups a functional interpretation.
5.3 Quotient groups
If N is normal in G, the set of cosets G/N can be made into a group. Quotient groups allow one to collapse a normal subgroup to the identity and study the resulting simplified structure.
5.3.1 Coset multiplication
The product of cosets is defined by multiplying representatives: (gN)(hN) = (gh)N. Normality is what guarantees that this operation does not depend on the chosen representatives. Thus the group law on the quotient is well defined only for normal subgroups.
5.3.2 First isomorphism theorem
The first isomorphism theorem states that the image of a homomorphism is isomorphic to the quotient of the domain by its kernel. This result shows how quotients naturally encode the effect of collapsing a normal subgroup. It is one of the central links between subgroups and structure-preserving maps.
6 Special classes of subgroups
Certain subgroups are distinguished by additional compatibility conditions or by their role in finite group structure. These classes often appear in deeper classification theorems.
6.1 Characteristic subgroups
A characteristic subgroup is invariant under every automorphism of the group. Because every automorphism is also an isomorphism from the group to itself, characteristic subgroups are stronger than normal subgroups. They are especially useful when studying subgroup behavior under internal symmetries.
6.2 Maximal subgroups
A maximal subgroup is a proper subgroup that is not properly contained in any other proper subgroup. Such subgroups sit just below the whole group in the inclusion order. They often help identify fundamental building blocks in finite and finitely generated groups.
6.3 Sylow subgroups
In finite group theory, Sylow subgroups are maximal subgroups whose order is a power of a prime. They encode the prime-power structure of a group and are central to the analysis of finite groups.
6.3.1 p-subgroups
A p-subgroup is a subgroup whose order is a power of a prime p. These subgroups reflect the p-local structure of a finite group. Their existence and arrangement give important information about the group as a whole.
6.3.2 Sylow theorems
The Sylow theorems describe when p-subgroups of a given size exist, how many there are, and how they are related by conjugation. These results are among the most powerful tools in finite group theory. They are widely used to prove structure and nonexistence statements.
7 Subgroups in specific group types
Different classes of groups display subgroup behavior in distinctive ways. Studying subgroups within these families often reveals patterns that are not visible in the abstract definition alone.
7.1 Subgroups of abelian groups
In abelian groups, every subgroup is automatically normal because conjugation is trivial. This simplifies the subgroup structure substantially and makes quotient constructions especially straightforward. Many classification results for abelian groups depend on understanding their subgroups.
7.2 Subgroups of cyclic groups
Cyclic groups have especially simple subgroup structure. Every subgroup of a cyclic group is cyclic, and finite cyclic groups have exactly one subgroup for each divisor of the group order. This makes cyclic groups a standard model for subgroup analysis.
7.3 Subgroups of finite groups
Finite groups have subgroup structures constrained by counting arguments and divisibility relations. The order of every subgroup divides the order of the group. This fact, together with Sylow theory, makes finite groups particularly amenable to systematic study.
7.4 Subgroups of permutation groups
Permutation groups often contain subgroups that preserve certain sets, blocks, or labels. Such subgroups can reflect combinatorial and geometric constraints. Because permutations encode symmetry so naturally, subgroup analysis here is closely tied to classification of actions.
8 Applications
Subgroups are not only central to pure group theory but also to many areas where symmetry and structure matter. They serve as a versatile language for decomposing and comparing algebraic systems.
8.1 Symmetry and geometry
In geometry, subgroups describe restricted sets of symmetries of a figure or space. For example, the symmetries preserving a chosen feature form a subgroup of the full symmetry group. This perspective helps classify geometric objects by their invariant properties.
8.2 Algebraic classification
Subgroups are used to analyze how groups are built from simpler components. Chains of subgroups, quotient constructions, and normal series all contribute to classification programs. Understanding subgroup structure is often the first step in understanding the entire group.
8.3 Galois theory
In Galois theory, subgroups of a Galois group correspond to intermediate field extensions. This relationship translates algebraic information about symmetries of roots into field-theoretic structure. Subgroups therefore provide the bridge between polynomial equations and field extensions.
8.4 Computational group theory
Algorithms for groups often rely on subgroup calculations such as membership testing, generation, coset enumeration, and index computation. These tasks are important in computer algebra systems and symbolic computation. Efficient handling of subgroups is essential for practical work with large or complex groups.