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
there exists an automorphism ϕ of X and an element such that
where "" is notation for an operation on X preserved by ϕ.
Two examples of algebraic structures with the property of closure with a twist are the cwatset and the GCset.
Contents
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 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, Sym(n), acts on Z_{2}^{n} by bit permutation:


 p((c_{1},...,c_{n}))=(c_{p(1)},...,c_{p(n)}),

where c=(c_{1},...,c_{n}) is an element of Z_{2}^{n} and p is an element of 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
This condition can also be written as
Examples
 All subgroups of Z_{2}^{n} — that is, nonempty subsets of Z_{2}^{n} which are closed under additionwithoutcarry — are trivially cwatsets, since we can choose each permutation p_{c} 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 01 matrix. For instance a matrix representation of F is given by
To see that F is a cwatset using this notation, note that
where π_{R} and σ_{C} denote permutations of the rows and columns of the matrix, respectively, expressed in cycle notation.
 For any another example of a cwatset is K_{n}, which has nbyn matrix representation
Note that for n = 3, K_{3} = F.
 An example of a nongroup cwatset with a rectangular matrix representation is
Properties
Let C Z_{2}^{n} be a cwatset.
 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
analogous to Lagrange's Theorem in group theory. To wit,
Theorem. If C is a cwatset of degree n and order m, then m divides 2^{n}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 (GCset) 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 GCset if for each h ∈ H, there exists a φ_{h} ∈ Aut(G) such that φ_{h}(h) H = φ_{h}(H).
Furthermore, a GCset H ⊆ G is a cyclic GCset 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} ϕ(h_{n − 1}) for all n > 1.
Examples
 Any cwatset is a GCset since C + c = π(C) implies that π ^{− 1}(c) + C = π ^{− 1}(C).
 Any group is a GCset, satisfying the definition with the identity automorphism.
 A nontrivial example of a GCset is H = {0, 2} where G = Z_{10}.
 A nonexample showing that the definition is not trivial for subsets of is H = {000, 100, 010, 001, 110}.
Properties
 A GCset H ⊆ G always contains the identity element of G.
 The direct product of GCsets is again a GCset.
 A subset H ⊆ G is a GCset if and only if it is the projection of a subgroup of Aut(G)⋉G, the semidirect product of Aut(G) and G.
 As a consequence of the previous property, GCsets have an analogue of Lagrange's Theorem: The order of a GCset divides the order of Aut(G)⋉G.
 If a GCset 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 subGCsets of orders p,p^{2},...,p^{q}. (Analogue of the first Sylow Theorem)
 A GCset is cyclic if and only if it is the projection of a cyclic subgroup of Aut(G)⋉G.
References
 Sherman, Gary J.; Wattenberg, Martin (1994), "Introducing … cwatsets!", Mathematics Magazine 67 (2): 109–117, doi:10.2307/2690684, JSTOR 2690684.
 The Cwatset of a Graph, NancyElizabeth Bush and Paul A. Isihara, Mathematics Magazine 74, #1 (February 2001), pp. 41–47.
 On the symmetry groups of hypergraphs of perfect cwatsets, Daniel K. Biss, Ars Combinatorica 56 (2000), pp. 271–288.
 Automorphic Subsets of the ndimensional Cube, Gareth Jones, Mikhail Klin, and Felix Lazebnik, Beiträge zur Algebra und Geometrie 41 (2000), #2, pp. 303–323.
 Daniel C. Smith (2003)RHITUMJ, RHIT [1]
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