Permutation groups

In MAGMA S8 := SymmetricGroup(8); (or Sym(8)) returns the symmetric group on $8$ 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 $S_n$, $n=3,4,5$, 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 $7$ letters. To find the number of elements of this group, type #S7;.

Exercise 5.20.3 (a)   Create the permutations group generated by $(1,2)$ and $(1,2,3,4,5,6,7)$, and find out how many elements they have.

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

Exercise 5.20.4   Create the subgroup of $S_{10}$ generated by $(1,2,3,4,5,6,7,8)$ and $(8,9,10)$. Compute $(1,2,3,4,5,6,7,8)^4*(8,9,10)^3$.

In MAGMA, to find the order of the move $R*F$ 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 $R*F$, $R*F*F*B$, $R*F*B*D*U*D$, in the Rubik's cube group^5.5.

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 $G$ be the group of symmetries of the square. Compute $G/\langle g_3\rangle$.

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 $G$ generated by $(1,\ 2,\ 3)$ and $(1,2)(3,4)$.

(b) List all the elements in the group $S_4$ generated by $(1,2)$ and $(1, 2, 3, 4)$.

(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 $D_4$ be the group geberated by $(1,2)(3,4)$, $(1, 2, 3, 4)$.

Exercise 5.20.11   Find all elements of order 2 in $D_4$.

Exercise 5.20.12   Find the conjugacy class of $(2,4)$ in the group $D_4$.

Exercise 5.20.13 (a)   Find the conjugacy classes of the group $S_3$.

(b) Find a complete set of representatives of each conjugacy class of $S_5$.

Exercise 5.20.14   Obtain both left and right coset representatives of $D_4$ in $S_4$.