Cosets are subsets of a mathematical group generated by a specific element. Only subgroups have cosets, and every number should be in exactly one coset. D4 is a noncommutative group with eight elements, and normal subgroups encode extra information.
A cosit is a specific type of subset of a mathematical group. For example, one could consider the set of all integer multiples of 7, {… -14, -7, 0, 7, 14 …}, which can be denoted as 7Z. Adding 3 to each number generates the set {… -11, -4, 3, 10, 17 …}, which mathematicians describe as 7Z + 3. This latter set is called a cosit of 7Z generated by 3.
There are two important properties of 7Z. If a number is a multiple of 7, so is its additive inverse. The additive inverse of 7 is -7, the additive inverse of 14 is -14, and so on. Also, adding a multiple of 7 to another multiple of 7 results in a multiple of 7. Mathematicians describe this by saying that multiples of 7 are “closed” in the operation of addition.
These two features are why 7Z is called a subgroup of addition integers. Only subgroups have cosets. The set of all cubic numbers, {… -27, -8, -1, 0, 1, 8, 27 …}, has no cosets in the same way as 7Z because it is not closed under addition: 1 + 8 = 9 and 9 is not a cubic number. Similarly, the set of all positive even numbers, {2, 4, 6, …}, has no cosits because it contains no inverse.
The reason for these clauses is that every number should be in exactly one coset. In the case of {2, 4, 6, …}, 6 is in the coset spawned by 4 and is in the cosit spawned by 2, but those two cosets are not identical. These two criteria are sufficient to ensure that each element is in exactly one object.
Cosets exist in any group, and some groups are much more complicated than integers. One useful group you might consider is the set of all the ways to move a square without changing the region it covers. If a square is rotated 90 degrees, there is no apparent change in shape. Likewise, it can be flipped vertically, horizontally, or across a diagonal without changing the region covering the square. Mathematicians call this group D4.
D4 has eight elements. Two elements are considered identical if they leave all corners in the same place, so rotating the square clockwise four times is considered to do nothing. With this in mind, the eight elements can be denoted by e, r, r2, r3, v, h, dd and dd. The “e” refers to doing nothing and the “r2” refers to performing two rotations. Each of the last four elements refers to the flipping of the square: vertically, horizontally or along its upward or downward sloping diagonals.
The integers are an abelian group, which means that its operation satisfies the commutative law: 3 + 2 = 2 + 3. D4 is not abelian. Rotating a square and then flipping it horizontally doesn’t move the corners in the same way as flipping it and then rotating it.
When working in noncommutative groups, mathematicians typically use a * to describe the operation. A little work shows that rotating the square and then flipping it horizontally, r * h, is equivalent to flipping it along its downward diagonal. So r * h = dd. Flipping the square and then rotating it is equivalent to flipping it along its diagonal up, so r * h = du.
Order is important in D4, so you need to be more precise when describing things. When working with integers, the phrase “the cosit of 7Z generated by 3” is unique because it doesn’t matter whether 3 is added to the left or right of each multiple of 7. For a subset of D4, however, different orders create different cosets. Based on the calculations described above, r*H, the left object of H generated by r—equals {r, dd} but H*r equals (r, du}. The requirement that no element be in two objects different does not apply when comparing right to left.
The right-hand elements of H do not match its left-hand elements. Not all subgroups of D4 share this property. We can consider the subgroup R of all rotations of the square, R={e, r, r2, r3}.
A little calculation shows that its left elements are the same as its right ones. Such a subgroup is called a normal subgroup. Normal subgroups are extremely important in abstract algebra because they always encode extra information. For example, the two possible things of R are equivalent to the two possible situations “the square has been flipped” and “the square has not been flipped”.
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