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The symmetric group on a set of size n is the Galois group of the general polynomial of degree n and plays an important role in Galois theory. In invariant theory, the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called symmetric functions.
The lattice of subgroups of the Symmetric group S 4, represented in a Hasse diagram. These are A005432 (4) = 30 distinct subgroups. They belong to A000638 (4) = 11 different types, which are shown on the right. S 4: Symmetric group of order 24 A 4: Alternating group of order 12 Dih 4: Dihedral group of order 8 S 3: Symmetric group of order 6
The A000638 (4) = 11 types of subgroups of the symmetric groups S 4 (Subgroups with the same colored cycle graph are bundled together.) Edge colors indicate the quotient of the connected groups' orders: red 2, green 3, blue 4
The dihedral group Dih 4 has ten subgroups, counting itself and the trivial subgroup. Five of the eight group elements generate subgroups of order two, and the other two non-identity elements both generate the same cyclic subgroup of order four. In addition, there are two subgroups of the form Z 2 × Z 2, generated by pairs of order-two ...
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids.
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The alternating group, symmetric group, and their double covers are related in this way, and have orthogonal representations and covering spin/pin representations in the corresponding diagram of orthogonal and spin/pin groups. Explicitly, S n acts on the n-dimensional space R n by permuting coordinates (in matrices, as permutation matrices).
This follows from inspection of 5-cycles: each 5-cycle generates a group of order 5 (thus a Sylow subgroup), there are 5!/5 = 120/5 = 24 5-cycles, yielding 6 subgroups (as each subgroup also includes the identity), and S n acts transitively by conjugation on the set of cycles of a given class, hence transitively by conjugation on these subgroups.