Symmetric group

Type of group in abstract algebra The symmetric group S 4 {\displaystyle \mathrm {S} _{4}} has 24 elements, viz. the permutations (depicted) of a 4-element set. A Cayley graph of the symmetric group S4 using the generators (red) a right circular shift of all four set elements, and (blue) a left circular shift of the first three set elements. 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} } ) Arith