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The Sylow subgroups of the symmetric groups are important examples of p-groups. They are more easily described in special cases first: The Sylow p-subgroups of the symmetric group of degree p are just the cyclic subgroups generated by p-cycles. There are (p − 1)!/(p − 1) = (p − 2)! such subgroups simply by counting generators.
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 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).
List of all nonabelian groups up to order 31 Order Id. [a] G o i Group Non-trivial proper subgroups [1] Cycle graph Properties 6 7 G 6 1: D 6 = S 3 = Z 3 ⋊ Z 2: Z 3, Z 2 (3) : Dihedral group, Dih 3, the smallest non-abelian group, symmetric group, smallest Frobenius group.
The full tetrahedral group T d with fundamental domain. T d, *332, [3,3] or 4 3m, of order 24 – achiral or full tetrahedral symmetry, also known as the (2,3,3) triangle group. This group has the same rotation axes as T, but with six mirror planes, each through two 3-fold axes. The 2-fold axes are now S 4 (4) axes.
The lattice theorem establishes a Galois connection between the lattice of subgroups of a group and that of its quotients. The Zassenhaus lemma gives an isomorphism between certain combinations of quotients and products in the lattice of subgroups. As groups are algebraic structures, it follows by a general Theorem (Burris & Sankappanavar 2011 ...
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