Quotient group

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 action Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) 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
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A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of ${\displaystyle n}$ and defining a group structure that operates on each such class (known as a congruence class) as a single entity. It is part of the mathematical field known as group theory.

For a congruence relation on a group, the equivalence class of the identity element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting quotient is written ⁠${\displaystyle G\,/\,N}$⁠, where ${\displaystyle G}$ is the original group and ${\displaystyle N}$ is the normal subgroup. This is read as '⁠${\displaystyle G{\bmod {N}}}$⁠', where ${\displaystyle {\mbox{mod}}}$ is short for modulo. (The notation ⁠${\displaystyle G\,/\,H}$⁠ should be interpreted with caution, as some authors (e.g., Vinberg^[1]) use it to represent the left cosets of ${\displaystyle H}$ in ${\displaystyle G}$ for any subgroup ${\displaystyle H}$, even though these cosets do not form a group if ${\displaystyle H}$ is not normal in ${\displaystyle G}$. Others (e.g., Dummit and Foote^[2]) only use this notation to refer to the quotient group, with the appearance of this notation implying the normality of ${\displaystyle H}$ in ${\displaystyle G}$.)

Much of the importance of quotient groups is derived from their relation to homomorphisms. The first isomorphism theorem states that the image of any group G under a homomorphism is always isomorphic to a quotient of ${\displaystyle G}$. Specifically, the image of ${\displaystyle G}$ under a homomorphism ${\displaystyle \varphi :G\rightarrow H}$ is isomorphic to ${\displaystyle G\,/\,\ker(\varphi )}$ where ${\displaystyle \ker(\varphi )}$ denotes the kernel of ⁠${\displaystyle \varphi }$⁠.

The dual notion of a quotient group is a subgroup, these being the two primary ways of forming a smaller group from a larger one. Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup. In category theory, quotient groups are examples of quotient objects, which are dual to subobjects.