Permutation groups

In GAP 4.1, s8 := SymmetricGroup(8); returns the GAP structure of the symmetric group on $8$ letters. To find the number of elements of this group, type Size( s8 );.

To find if it is an abelian group, type IsAbelian( s8 );.

Exercise 5.19.2   Create the symmetric group $S_n$, $n=3,4,5$, and find out how many elements they have.

In GAP, a permutation group can be entered using the Group command. For example, s7 := Group( (1,2), (1,2,3,4,5,6,7) ); returns the GAP structure of the group generated by $(1,2)$ and $(1,2,3,4,5,6,7)$, which happens to be the symmetric group on $7$ letters. To find the number of elements of this group, type Size( s7 );.

Exercise 5.19.3 (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 5.19.4   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 GAP, to find the order of the move $R*F$ of the Rubik's cube, type

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

Exercise 5.19.5   Find the orders of $R*F$, $R*F*F*B$, $R*F*B*D*U*D$, in the Rubik's cube group^5.4.

The Rotation game. Recall the $3\times 3$ grid,

$$\begin{array}{ccc} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \end{array},$$

from §4.6. The allowed moves are rotations of the following form, or combinations thereof: In cycle notation, $r_1=(a_1,a_2 ,a_5 ,a_4 )$, $r_2=(a_2 ,a_3 ,a_6 ,a_5 )$, $r_3=(a_4 ,a_5 ,a_8 ,a_9 )$, or $r_4=(a_5 ,a_6 ,a_9 ,a_8 )$. In other words, a legal move in the Rotation game is any permutation of the form $r_{i_1}r_{i_2}...r_{i_k}$, where $1\leq i_j\leq 4$ for all $1\leq j\leq N$.

Question: Can each permutation of $\{1,2,...,9\}$ be expressed in the form $r_{i_1}r_{i_2}...r_{i_k}$, where $1\leq i_j\leq 4$ for all $1\leq j\leq N$?

Exercise 5.19.6   Answer this question.

In GAP,

G:=Group((1,2,3), (1,2));
c:=RightCoset(G,(2,3,4)); 
Elements(c);

Exercise 5.19.7   Let $G$ be the group of symmetries of the square. Compute $G/\langle g_3\rangle$.

Exercise 5.19.8 (a)   Find the sign and swapping number of (1,4,7)(2,5).

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

Exercise 5.19.9 (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 5.19.10 (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 5.19.11   Find all elements of order 2 in $D_4$.

Exercise 5.19.12   Find the conjugacy class of $(2,4)$ in the group $D_4$.

Exercise 5.19.13 (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 5.19.14   Obtain both left and right coset representatives of $D_4$ in $S_4$.

Exercise 5.19.15   Answer the question about the Rotation game in §4.6 using GAP.