Closure with a twist

Closure with a twist is a property of subsets of an algebraic structure. A subset $$Y$$ of an algebraic structure $$X$$ is said to exhibit closure with a twist if for every two elements


 * $$ y_1, y_2 \in Y$$

there exists an automorphism $$\phi$$ of $$X$$ and an element $$y_3 \in Y$$ such that
 * $$ y_1 \cdot y_2 = \phi(y_3)$$

where "$$\cdot$$" is notation for an operation on $$X$$ preserved by $$\phi$$.

Two examples of algebraic structures which exhibit closure with a twist are the cwatset and the generalized cwatset, or GC-set.

Cwatset
In mathematics, a cwatset is a set of bitstrings, all of the same length, which is closed with a twist.

If each string in a cwatset, C, say, is of length n, then C will be a subset of $$\mathbb{Z}_2^n$$. Thus, two strings in C are added by adding the bits in the strings modulo 2 (that is, addition without carry, or exclusive disjunction). The symmetric group on n letters, $$\text{Sym}(n)$$, acts on $$\mathbb{Z}_2^n$$ by bit permutation:
 * $$p((c_1, \ldots, c_n)) = (c_{p(1)}, \ldots, c_{p(n)}),$$

where $$c = (c_1, \ldots, c_n)$$ is an element of $$\mathbb{Z}_2^n$$ and p is an element of $$\text{Sym}(n)$$. Closure with a twist now means that for each element c in C, there exists some permutation $$p_c$$ such that, when you add c to an arbitrary element e in the cwatset and then apply the permutation, the result will also be an element of C. That is, denoting addition without carry by $$+$$, C will be a cwatset if and only if
 * $$\forall c\in C : \exists p_c\in \text{Sym}(n) : \forall e\in C : p_c(e+c) \in C.$$

This condition can also be written as
 * $$\forall c\in C : \exists p_c\in \text{Sym}(n) : p_c(C+c)=C.$$

Examples

 * All subgroups of $$\mathbb{Z}_2^n$$ &mdash; that is, nonempty subsets of $$\mathbb{Z}_2^n$$ which are closed under addition-without-carry &mdash; are trivially cwatsets, since we can choose each permutation pc to be the identity permutation.
 * An example of a cwatset which is not a group is


 * F = {000,110,101}.

To demonstrate that F is a cwatset, observe that
 * F + 000 = F.
 * F + 110 = {110,000,011}, which is F with the first two bits of each string transposed.
 * F + 101 = {101,011,000}, which is the same as F after exchanging the first and third bits in each string.


 * A matrix representation of a cwatset is formed by writing its words as the rows of a 0-1 matrix. For instance a matrix representation of F is given by


 * $$ F = \begin{bmatrix}

0 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}.$$

To see that F is a cwatset using this notation, note that


 * $$ F + 000 = \begin{bmatrix}

0 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} = F^{id}=F^{(2,3)_R(2,3)_C}. $$


 * $$ F + 110 = \begin{bmatrix}

1 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 1 \end{bmatrix} = F^{(1,2)_R(1,2)_C}=F^{(1,2,3)_R(1,2,3)_C}. $$


 * $$ F + 101 = \begin{bmatrix}

1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix} = F^{(1,3)_R(1,3)_C}=F^{(1,3,2)_R(1,3,2)_C}.$$

where $$ \pi_R$$ and $$ \sigma_C$$ denote permutations of the rows and columns of the matrix, respectively, expressed in cycle notation.


 * For any $$ n \geq 3 $$ another example of a cwatset is $$ K_n $$, which has $$n$$-by-$$n$$ matrix representation


 * $$ K_n = \begin{bmatrix}

0 & 0 & 0 & \cdots & 0 & 0 \\ 1 & 1 & 0 & \cdots & 0 & 0 \\ 1 & 0 & 1 & \cdots & 0 & 0 \\ & & & \vdots & & \\ 1 & 0 & 0 & \cdots & 1 & 0 \\ 1 & 0 & 0 & \cdots & 0 & 1 \end{bmatrix}. $$

Note that for $$ n = 3$$, $$K_3=F$$.


 * An example of a nongroup cwatset with a rectangular matrix representation is


 * $$ W = \begin{bmatrix}

0 & 0 & 0\\ 1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1 \end{bmatrix}. $$

Properties
Let $$C \subset \mathbb{Z}_2^n$$ be a cwatset.

analogous to Lagrange's Theorem in group theory. To wit,
 * The degree of C is equal to the exponent n.
 * The order of C, denoted by |C|, is the set cardinality of C.
 * There is a necessary condition on the order of a cwatset in terms of its degree, which is

Theorem. If C is a cwatset of degree n and order m, then m divides $$2^n!$$.

The divisibility condition is necessary but not sufficient. For example, there does not exist a cwatset of degree 5 and order 15.

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

Definitions
A subset H of a group G is a GC-set if for each $$h\in H$$, there exists a $$\phi_h\in\text{Aut}(G)$$ such that $$\phi_h(h) \cdot H = \phi_h(H)$$.

Furthermore, a GC-set H ⊆ G is a cyclic GC-set if there exists an $$h\in H$$ and a $$\phi\in\text{Aut}(G)$$ such that $$H = {h_1, h_2, ...}$$ where $$h_1 = h$$ and $$h_n = h_1 \cdot \phi(h_{n-1})$$ for all $$n > 1$$.

Examples

 * Any cwatset is a GC-set, since $$C + c = \pi(C)$$ implies that $$\pi^{-1}(c) + C = \pi^{-1}(C)$$.
 * 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 $$Z_2^n$$ is $$H = {000, 100, 010, 001, 110}$$.

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.