Closure with a twist

Closure with a twist is a property of subsets of an algebraic structure. A subset formula_1 of an algebraic structure formula_2 is said to exhibit closure with a twist if for every two elements
there exists an automorphism formula_4 of formula_2 and an element formula_6 such that 
where "formula_8" is notation for an operation on formula_2 preserved by formula_4.
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 formula_11. 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, formula_12, acts on formula_11 by bit permutation:
where formula_15 is an element of formula_11 and "p" is an element of formula_12. Closure "with a twist" now means that for each element "c" in "C", there exists some permutation formula_18 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 formula_19, "C" will be a cwatset if and only if
This condition can also be written as
Examples.
To demonstrate that "F" is a cwatset, observe that
To see that "F" is a cwatset using this notation, note that
where formula_28 and formula_29 denote permutations of the rows and columns of the matrix, respectively, expressed in cycle notation.
Note that for formula_35, formula_36.
Properties.
Let formula_38 be a cwatset.
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 formula_39.
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 formula_40, there exists a formula_41 such that formula_42.
Furthermore, a GC-set "H" ⊆ "G" is a "cyclic GC-set" if there exists an formula_40 and a formula_44 such that formula_45 where formula_46 and formula_47 for all formula_48.