Infinite group

In today's world, Infinite group has become a topic of increasing interest. Whether due to its impact on society, its relevance in the academic field or its influence on popular culture, Infinite group has positioned itself as a topic of constant debate and reflection. From its origins to the present, Infinite group has been the subject of study and analysis by experts in different areas, who have tried to decipher its multiple facets and understand its importance in the development of humanity. In this article, we will explore the different perspectives from which Infinite group can be approached, and we will analyze its relevance in the contemporary world.

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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In group theory, an area of mathematics, an infinite group is a group whose underlying set contains an infinite number of elements. In other words, it is a group of infinite order.

Examples

• (Z, +), the group of integers with addition is infinite
• Non-discrete Lie groups are infinite. For example, (R, +), the group of real numbers with addition is an infinite group
• The general linear group of order n > 0 over an infinite field is infinite

See also

Finite group

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