Subgroup
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| Subgroup | |
|---|---|
| Name | Subgroup |
| Field | Group theory |
| Studied in | Abstract algebra |
| Examples | Cyclic group, Alternating group, Symmetric groups, Dihedral group, Matrix group |
| Key properties | Closure, identity, inverses, associativity inherited |
| Related concepts | Normal subgroup, Quotient group, Group action, Group homomorphism |
Subgroup A subgroup is a subset of a group that itself forms a group under the same binary operation. Subgroups appear throughout Abstract algebra, linking concrete examples such as (\u211a,+), finite groups like Alternating groups and symmetric groups, and structural results such as the Jordan–Hölder theorem and the Sylow theorems. They provide the language for constructing quotient groups, analyzing homomorphisms, and classifying groups via series and lattice-theoretic methods.
Definition and basic properties
A subset H of a group G is a subgroup if H contains the identity element of G, is closed under the group operation, and contains inverses for all its elements; associativity is inherited from G. Basic consequences include that the intersection of any collection of subgroups of G is again a subgroup, and that every element g in G generates a cyclic subgroup
Examples and classification
Canonical examples: for n≥2 the Symmetric group S_n contains the Alternating group A_n as a subgroup of index 2; Dihedral group D_n contains rotation subgroups isomorphic to Cyclic groups and reflection-generated subgroups. In matrix theory, General linear group GL(n, K) contains Special linear group SL(n, K) and various block-diagonal matrix subgroups; Orthogonal group O(n) and Unitary group U(n) are subgroups of GL(n, R) and GL(n, C) respectively. Arithmetic and topology offer examples: the group of integer translations Z in the real line R under addition, the Modular group PSL(2, Z) with congruence subgroups such as Γ_0(N), and fundamental-group subgroups arising from covering spaces in topology linked to the Seifert–van Kampen theorem. Classification results include the lattice of subgroups of finite cyclic groups, the subgroup structure of p-groups governed by Burnside problem-related constraints, and deep classifications for simple groups such as Monster group where maximal subgroups were determined by the ATLAS project.
Subgroup tests and criteria
Practical criteria determine when a subset H ≤ G. The one-step subgroup test: nonempty H closed under the product ab^{-1} implies H is a subgroup. The two-step test: H nonempty closed under multiplication and inverses suffices. For finite G, closure under multiplication alone implies closure under inverses, simplifying checks for subgroups of Permutation groups and Matrix groups. Other specific criteria include tests for cyclicity (existence of a generator), Sylow criteria from the Sylow theorems for p-subgroups, and results for Hall subgroup existence in solvable groups via Frobenius theorem and Schur–Zassenhaus theorem when orders satisfy coprimality conditions.
Normal subgroups and quotient groups
A subgroup N of G is normal if gNg^{-1}=N for all g in G; normality is equivalent to left and right cosets coinciding and to kernel characterization: kernels of homomorphisms are normal subgroups. Normal subgroups permit construction of Quotient groups G/N with the canonical projection map; fundamental results include the First isomorphism theorem, Second isomorphism theorem, and Third isomorphism theorem linking quotients, subgroups, and images. Simple groups are nontrivial groups whose only normal subgroups are trivial and whole, a concept central to the Classification of Finite Simple Groups and illustrated by families like A_n for n≥5 and various Lie group analogues. Extensions and cohomology theories, including Group cohomology and group extensions, study how a group can be built from a normal subgroup and a quotient.
Subgroup lattices and structure theorems
The set of all subgroups of G, ordered by inclusion, forms the subgroup lattice L(G), a complete lattice where meet is intersection and join is subgroup generated by a union. Modular and distributive properties of L(G) reflect group structure; for example, Dedekind groups have normality of all subgroups, and Jordan–Hölder theorem gives unique composition factors up to permutation for finite groups, reflected in maximal chains of L(G). The study of maximal subgroups, minimal normal subgroups, and composition series connects to results like Schreier refinement theorem and Frattini subgroup Φ(G), which is the intersection of all maximal subgroups and governs generators. Lattice isomorphism problems relate to Hölder's theorem and investigations in the ATLAS project for concrete finite groups.
Applications and related concepts
Subgroups underpin symmetry analysis in Chemistry, Crystallography, and Physics via point groups and space groups such as those cataloged in crystallographic tables; they organize conservation laws in Particle physics through subgroups of Lie groups like SU(3), SU(2), and U(1). In number theory, congruence subgroups of Modular groups influence modular forms and the Modularity theorem; in geometry, deck transformation groups and covering-space subgroups classify coverings via Fundamental group correspondences. Computational group theory tools—GAP, Magma—exploit subgroup algorithms for coset enumeration, Sylow subgroup finding, and subgroup lattice computations. Related notions include Normal closure, core of a subgroup, Subgroup growth, and subgroup generation problems central to modern research in Geometric group theory and finite-group classification.