Subgroup

In group theory, a branch of mathematics, given a group ${\displaystyle G}$ under a binary operation, a subset ${\displaystyle H}$ of ${\displaystyle G}$ is called a subgroup of ${\displaystyle G}$ if the elements ${\displaystyle H}$ also forms a group under the binary operation.

Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable List of group theory topics
Finite groups Classification of finite simple groups cyclic alternating Lie type sporadic Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Symmetric group Sn Klein four-group V Dihedral group Dn Quaternion group Q Dicyclic group Dicn
Discrete groups Lattices Integers (Z) Free group Modular groups PSL(2,Z) SL(2,Z) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve

The situation could arise in which the subset ${\displaystyle H}$ does not comply with the binary operation.