## Permutation groups

In MAGMA S8 := SymmetricGroup(8); (or Sym(8)) returns the symmetric group on letters. To find the number of elements of this group, type #S8;.

To find if it is an abelian group, type IsAbelian( S8 );.

Exercise 5.20.2   Create the symmetric group , , and find out how many elements they have.

In MAGMA, a permutation group can be entered using the PermutationGroup command. For example,
S7 := PermutationGroup<7 | (1,2), (1,2,3,4,5,6,7)> ; returns the symmetric group on letters. To find the number of elements of this group, type #S7;.

Exercise 5.20.3 (a)   Create the permutations group generated by and , and find out how many elements they have.

(b) Create the permutations group generated by in the section discussing Plain Bob Minimus and find out how many elements they have.

Exercise 5.20.4   Create the subgroup of generated by and . Compute .

In MAGMA, to find the order of the move of the Rubik's cube, type

S48:=Sym(48);
U:=S48!( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19);
L:= S48!( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35);
F:=S48!(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11);
R:=S48!(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24);
B:=S48!(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27);
D:=S48!(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40);
G:=PermutationGroup<48!U,L,F,R,B,D>;
Order(R*F);


Exercise 5.20.5   Find the orders of , , , in the Rubik's cube group5.5.

Exercise 5.20.6   Answer the question about the Rotation game in §4.6 using MAGMA.

In MAGMA,

H:=Sym(4);
G:=PermutationGroup<4 | (1,2,3),(1,2)>;
ElementSet(G, G);
h:=H!(2,3,4);
c:=G*h;


Exercise 5.20.7   Let be the group of symmetries of the square. Compute .

Sign(g) returns 1 if the permutation g is even, return -1 if g is odd.

Exercise 5.20.8 (a)   Find the sign and swapping number of (1,4,7)(2,5).

(b) Same for the Rubik's cube move B.

Exercise 5.20.9 (a)   Multiply (1,2,3,4,5,6,7) times (2,5) times (7,6,5,4,3,2,1).

(b) Multiply R*L*U*DD*F*B.

Exercise 5.20.10 (a)   List all the elements in the group generated by and .

(b) List all the elements in the group generated by and .

(c) How many positions of the Rubik's cube can be obtained by only using the moves R and F? (Hint: Consider the "two faces subgroup" of the Rubik's cube group generated by F and R.)

Let be the group geberated by , .

Exercise 5.20.11   Find all elements of order 2 in .

Exercise 5.20.12   Find the conjugacy class of in the group .

Exercise 5.20.13 (a)   Find the conjugacy classes of the group .

(b) Find a complete set of representatives of each conjugacy class of .

Exercise 5.20.14   Obtain both left and right coset representatives of in .

david joyner 2008-04-20