So, permutation matrices do indeed permute the order of elements in vectors multiplied with them. Permuting the vertices causes the factorizations to permute differently, in accordance with the outer automorphism. A new set of permute instructions have been added for full two input permutations. Galois theory concerns transformations of number fields that permute the roots of an equation. Circular shifts are used often in cryptography in order to permute bit sequences. A "computation medium" is a system whose attributes permute to always produce a possible task. In particular, since normal subgroups permute with each other, they form a modular sublattice. This is independent of "k" (simply permute points). It can be twisted along its cuts to permute its pieces.