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Transformations induced by a mathematical group Algebraic structure → Group theoryGroup theory Basic notions Subgroup Normal subgroup Group action Quotient group (Semi-)direct product Direct sum Free product Wreath product Group homomorphisms kernel image simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable 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 groupsLattices Integers ( Z {\displaystyle \mathbb {Z} } ) Free group Modular groups PSL(2, Z {\displaystyle \mathbb {Z} } )SL(2, Z {\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 vte The cyclic group C3 consisting of the rotations by 0°, 120° and 240° acts on the set of the three vertices.

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