subdegrees

In the study of group theory, specifically within the context of permutation groups, the concept of subdegrees arises. When a permutation group G acts on a set X, it partitions X into orbits. If the group is transitive, meaning it acts on only one orbit (the entire set X), we can examine further substructures. The subdegrees of a transitive group G are related to the number of points fixed by the stabilizers of elements within the group.

More formally, let G be a transitive permutation group acting on a set X. For any element x in X, let G_x be the stabilizer of x, meaning the set of group elements that leave x fixed. The group G acts on the set G/G_x, which is the set of right cosets of G_x in G. This action is also transitive. The orbits of the action of G_x on X, excluding the orbit containing x itself, are related to the subdegrees.

The subdegrees are the sizes of these G_x-orbits on X \ {x}. If we choose a point x, and consider the action of its stabilizer G_x on the remaining points X \ {x}, this action might decompose X \ {x} into several orbits. The number of points in each of these orbits are the subdegrees of the group G. For a transitive group, there is always at least one subdegree, which is the size of the orbit of G_x on X, which is |X| - 1 (the number of points not fixed by G_x). The subdegrees partition the set X \ {x} in a specific way, and their sum, along with the fixed point (which can be thought of as an orbit of size 1 under the trivial action of G_x), equals the total number of points in X. The subdegrees provide important information about the structure of transitive permutation groups and are used in their classification and analysis.