GC-set

From Wikipedia, the free encyclopedia

In mathematics, generalized cwatset (GC-set) is an algebraic structure generalizing the notion of closure with a twist, the defining characteristic of the cwatset.

1 Definitions
2 Examples
3 Properties
4 See also

[edit] Definitions

A subset H of a group G is a GC-set if for each h ∈ H, there exists a $φ h$ ∈ Aut(G) such that $φ h$ (h) $\cdot$ H = $φ h$ (H).

Furthermore, a GC-set H ⊆ G is a cyclic GC-set if there exists an h ∈ H and a $φ$ ∈ Aut(G) such that H = { $h 1, h 2,...$ } where $h 1$ = h and $h n$ = $h 1$ $\cdot$ $φ$ ( $h n - 1$ ) for all n > 1.

[edit] Examples

Any cwatset is a GC-set since C + c = $π$ (C) implies that $π - 1$ (c) + C = $π - 1$ (C).
Any group is a GC-set, satisfying the definition with the identity automorphism.
A non-trivial example of a GC-set is: H = {0, 2} where G = $Z 10$ .
A NONEXAMPLE (showing that the definition is not trivial for subsets of $Z_2^n$ ): H = {000, 100, 010, 001, 110}.

[edit] Properties

A GC-set H ⊆ G always contains the identity element of G.
The direct product of GC-sets is again a GC-set.
A subset H ⊆ G is a GC-set if and only if it is the projection of a subgroup of Aut(G)⋉G, the semi-direct product of Aut(G) and G.
As a consequence of the previous property, GC-sets have an analogue of Lagrange's Theorem: The order of a GC-set divides the order of Aut(G)⋉G.
If a GC-set H has the same order as the subgroup of Aut(G)⋉G of which it is the projection then for each prime power $p q$ which divides the order of H, H contains sub-GC-sets of orders p, $p 2$ ,..., $p q$ . (Analogue of the first Sylow Theorem)
A GC-set is cyclic if and only if it is the projection of a cyclic subgroup of Aut(G)⋉G.