Group Theory for Puzzles 07/08/2007 04:27 PM

Group Theory for puzzles 07/08/2007 04:27 PM Group Theory for puzzles On this page I will give much more detail on what groups are, filling in many details that are skimmed over on the Useful Mathematics page. However, since there is a lot to deal with, I will go through it fairly quickly and not elaborate much, which makes the text rather dense. I will at the same time try to relate these concepts to permutation puzzles, and the Rubik's Cube in particular. 1. Sets and functions 2. Groups and homomorphisms 3. Actions and orbits 4. Cosets and coset spaces 5. Permutations and parity 6. Symmetry and Platonic solids 7. Conjugation and commutation 8. Normal groups and quotient groups 9. Bibliography and Further reading Algebra and Group Theory Application to puzzles 1. Sets and functions A set is a collection of things called elements, and each of these elements occurs at most once in a set. A (finite) set can be given explicitly as a list inside a pair of curly brackets, for example {2,4,6,8} is the set of even positive integers below 10. It has four elements. There are infinite sets - sets that have an infinite number of elements - such as the set of integers, but I On puzzles, we will often consider the set of puzzle will generally discuss only finite sets (and groups) here. positions. A function f:A B from set A to set B is a pairing that assigns to each element in A some element of B. Set A is called the A move on a puzzle is usually a function on the set domain and set B the range. If the function f assigns element of positions. A move applied to one position results b to a then we say that f maps a to b, and call b the image of a. in another position. This assumes however that a We could write this as f(a)=b, but I prefer to write it as af=b. move can always be applied to any position - in Writing f on the right like this will also be more natural in the some puzzles such as the Square-1 this is not the cube context later, but be aware that in many books it is written case and makes such a puzzle much harder to on the left instead. analyse. A function is injective or one-to-one if the function maps different elements in A to different elements in B. A function is surjective or onto if every element in B is mapped to by the function. A function is called a bijection if it is both one-to-one and onto. http://www.geocities.com/jaapsch/puzzles/groups.htm Page 1 of 18 Group Theory for puzzles 07/08/2007 04:27 PM Lemma 1.1: For a bijection f:A B we can uniquely define a function g:B A, such that if a is any element of A and b is its image in B under f (i.e. af=b) then bg=a. Proof: Let b be an element of B. Since f is onto, there must be some a in A such that af=b. Since f is one-to-one, this element a is uniquely determined. Therefore it is actually possible to simply define g as that pairing that maps b to a whenever f maps a to b. The function above is called the inverse function of f, and is usually denoted by f -1. Thus every bijection f:A B has an inverse f -1:B A such that whenever af=b for some a in A and b in B, then bf -1=a. Lemma 1.2: The inverse of a bijection is also a bijection. Proof: Let f:A B be a bijection, and f -1:B A its inverse. Let a be any element in A, and let b=af. Then a=bf -1. Therefore f -1 is surjective. Again let a be any element of A. There is a unique b in B such that af=b. In other words there is a unique b in B On nearly all puzzles, moves have inverses. A such that bf -1=a. Thus f -1 is one-to-one. move can usually be undone, taking you back to the position you had before. For example on the cube, Note that if f:A B is a bijection, then f uniquely pairs each the move R, a clockwise turn of the right face, can element of A with an element of B and vice versa. If A and B be undone by an anti-clockwise turn of the right and finite, they will therefore have the same number of face. This can be denoted by R-1, the inverse of R, elements. though R' is more common. The composition of two functions f:A B and g:B C, is the function that is the result of applying first f and then g. Thus to an element a in A we map the element c in C given by c=(af)g. This function is denoted f g. Lemma 1.3: If f:A B and g:B C are one-to-one, then f g is also one-to-one. Proof: Suppose a1 and a2 are elements of A, such that a1(f g)=a2(f g). Then (a1f)g=(a2f)g and since g is one-to-one we must have a1f=a2f. Since f is one-to-one, we get a1=a2. Moves can be combined into move sequences. Therefore f g never maps different elements to the same image Instead of having to keep track of what each move element, in other words it is also one-to-one. in a move sequence does to a position, you instead think only of what the sequence as a whole does to Lemma 1.4: If f:A B and g:B C are onto, then f g is also onto. it. A move sequence has become a single entity in Proof: Let c be an arbitrary element of C. There must be some itself, which is applied to a position. element b in B such that bg=c since g is onto. There must be an a in A such that af=b since f is onto. Therefore Any move sequence is a composition of functions. c=bg=(af)g=a(f g) which means that f g is onto. For example, if we apply the move sequence FRB to a position p, we can do the moves one by one, Lemma 1.5: If f:A B and g:B C are bijections, then f g is also which gives position ((pF)R)B, or we can consider it a bijection. as a single function F R B, which gives position Proof: Follows directly from previous two lemmas. p(F R B). The identity function on A is the function iA:A A defined by Note that if two different move sequences always ai=a for every a in A. It is easily seen to be a bijection. have the same effect on the cube, then they are simply different ways of writing down the same function. Most of the theory developed here is Lemma 1.6: If f:A B is a function, then f iB=iA f=f. Proof: For any a in A, we have a(f i )=(af)i =af=(ai )f=a(i f). concerned only with the effect, not with how the B B A A function is represented by a move sequence. The three functions f, f iB, and iA f all have the same effect on http://www.geocities.com/jaapsch/puzzles/groups.htm Page 2 of 18 Group Theory for puzzles 07/08/2007 04:27 PM any element, they must therefore be the same function. The identity on the cube is the move sequence that contains no moves at all, or any other move -1 -1 Lemma 1.7: If f:A B is a bijection, then f f =iA and f f=iB. sequence that does nothing such as Proof: For any a in A, we have an element b in B such that F2B2L2R2F2B2L2R2. -1 -1 -1 -1 b=af. Then bf =a which gives a(f f )=(af)f =bf =a=aiA. Therefore f f -1=i . Conversely for any b in B we can find a in The move sequences U2D2 R'L FB' and A RL'FB'RL'F'B are inverses. You can check this by -1 -1 -1 A such that bf =a and then b(f f)=(bf )f=af=b. Therefore doing the first and then the second sequence on a -1 f f=iB. solved cube to again find the cube solved. This automatically means that the second sequence followed by the first will also do nothing to the cube. Lemma 1.8: Composition is associative, i.e. for functions f, g, and h we have f (g h)=(f g) h. Proof: Let a be any element of the set on which f is defined, then a(f (g h))=(af)(g h))=((af)g)h=(a(f g))h=a((f g) h). Thus f (g h) and (f g) h have the same effect on any element, and are therefore the same function. To undo or invert the move sequence FRB, we Lemma 1.9: (f g)-1=g-1 f -1.

Load more