Let $p_1, p_2, p_3, \ldots$ be the sequence of prime integers, and set $E_n = p_1 p_2 \cdots p_n + 1$.
Compute the values $E_n$ for $n = 1, 2, 3, \ldots$; which of them are prime integers?
Let $G$ be the group generated by the permutations $(1,2,3,…,n)$ and $(1,p)$, for some $p$. Try several values of $n$ and $p$: When is $G$ the full symmetric group on $n$ points?
Replace $(1,p)$ by $(1,p,q)$.
Let $G$ be the group generated by the permutations $(1,3,4,…,n)$ and $(1,2,3)$. Try several values of $n$: When is $G$ the full symmetric group on $n$ points?
Replace $(1,2,3)$ by $(1,2,3,4)$.
Show that the groups SL(2,4) and $A_5$ are isomorphic.
Consider a cube whose vertices are labeled as follows.
2 ___________ 1 _ y
/| /| /|
/ | 4 / | /
3 /__|_______/ | -/----> x
| | | |
| |_______|__| 7
| / 8 | /
|/ |/
/__________/
5 6
Rotations around the axes x and y
induce the permutations (1,4,6,7)(2,3,5,8)
and (1,7,8,2)(4,6,5,3), respectively,
of the vertices.
Let $G$ be the group generated by these permutations.
The action of $G$ on the edges of the cube can be constructed as the action
on the orbit of the set [ 1, 2 ] via OnSets.
Consider the actions of $G$ on the faces, face diagonals, diagonals, pairs of opposite edges, and pairs of opposite faces, with appropriate action functions in GAP.
In each case, compute orbit length, point stabilizer, and the order of the image of the action homomorphism.
What does the action on the orbit of [ 1, 3, 6, 8 ]
via OnSets describe?
For $n = 2, 3, 4, 5$,
compute a faithful permutation representation of the group
$\langle x, y; x^2 = y^3 = (x y)^n = 1 \rangle$,
determine the group orders;
try also StructureDescription.
The file rubik2.g contains GAP input for a permutation representation of the group of moves of Rubik’s $2 \times 2 \times 2$ cube, on $21$ points.
(Download the file and call Read("rubik2.g")
in your GAP session.)
Study the structure of this permutation group, similar to the study of Rubik’s $3 \times 3 \times 3$ cube in https://www.math.rwth-aachen.de/homes/GAP/WWW2/Doc/Examples/rubik.html:
Factorization for solving
the puzzle.Find a group $G$ of order $2^9$ such that
One possibility is to use the classification of groups of order $2^9$:
SmallGroupsInformation( 2^9 ) shows ranks and $p$-classes of the
groups of order $2^9$.
To which ranges can the group $G$ belong?
(Such a group is constructed by hand in Section 3 of a paper by B. Brewster and G. Yeh (J. Algebra 146 (1992), 18-29).)
Let $A_5$ denote the alternating group on five points. Consider the direct product of $n$ copies of $A_5$, for $n = 2, 3, \ldots$.
Show that for small $n$, it is possible to find a generating set for this direct product that consists of two elements.
Background:
The derived subgroup G’ of a group G is defined as the subgroup
that is generated by the
Find a group G with the property that not every element in G’ is a commutator.
For a given positive integer n,
the ring $R$ of residue classes of integers modulo n
can be created with Integers mod n.
For different values of n, compute
The group of invertible $k \times k$ matrices over $R$ can be created as GL$( k, R )$. Compute the order of this group, for various values of $n$, for example increasing powers of a prime.
Let l be a list of pairs,
for example
l = [ [ 1, 1 ], [ 2, 1 ], [ 1, 2 ], [ 2, 2 ], [ 2, 3 ], [ 3, 1 ] ],
and suppose that the task is to compute the sublist of those pairs
that are the first ones with given first entry.
In the above example, the result would be
[ [ 1, 1 ], [ 2, 1 ], [ 3, 1 ] ].
An elegant way to solve this task would be
Filtered( l, x -> x = First( l, y -> y[1] = x[1] ) ).
Is this approach a good idea?
Compare this approach with a solution via a for loop,
for different lengths of l.
Consider the cube in Exercise G4, embedded into the 3-dimensional space such that all coordinates of the vertices have absolute value 1.
Then the symmetries can be represented by integral matrices.
Write down the matrices corresponding to the above permutations.
Show that mapping the permutations to the matrices
really defines a group isomorphism,
using the function GroupHomomorphismByImages
or via the action of the matrix group on the orbit of the row vector
[ 1, 1, 1 ].
Construct the various actions mentioned in Exercise G4 (on vertices, edges, faces, etc.) as actions of the matrix group on orbits of suitable sets of row vectors.
Let $p_1, p_2, p_3, \ldots$ be the sequence of prime integers, and set $E_n = p_1 p_2 \cdots p_n + 1$.
Compute the values $E_n$ for $n = 1, 2, 3, \ldots$; which of them are prime integers?
Let $G$ be the group generated by the permutations $(1,2,3,…,n)$ and $(1,p)$, for some $p$. Try several values of $n$ and $p$: When is $G$ the full symmetric group on $n$ points?
Replace $(1,p)$ by $(1,p,q)$.
Let $G$ be the group generated by the permutations $(1,3,4,…,n)$ and $(1,2,3)$. Try several values of $n$: When is $G$ the full symmetric group on $n$ points?
Replace $(1,2,3)$ by $(1,2,3,4)$.
Show that the groups SL(2,4) and $A_5$ are isomorphic.
Consider a cube whose vertices are labeled as follows.
2 ___________ 1 _ y
/| /| /|
/ | 4 / | /
3 /__|_______/ | -/----> x
| | | |
| |_______|__| 7
| / 8 | /
|/ |/
/__________/
5 6
Rotations around the axes x and y
induce the permutations (1,4,6,7)(2,3,5,8)
and (1,7,8,2)(4,6,5,3), respectively,
of the vertices.
Let $G$ be the group generated by these permutations.
The action of $G$ on the edges of the cube can be constructed as the action
on the orbit of the set Set([1, 2]).
Consider the actions of $G$ on the faces, face diagonals, diagonals, pairs of opposite edges, and pairs of opposite faces, by constructing orbits of suitable objects.
In each case, compute orbit length, point stabilizer, and the order of the image of the action homomorphism.
What does the action on the orbit of Set([1, 3, 6, 8]) describe?
For $n = 2, 3, 4, 5$,
compute a faithful permutation representation of the group
$\langle x, y; x^2 = y^3 = (x y)^n = 1 \rangle$,
determine the group orders;
try also describe.
The file rubik2.jl contains OSCAR input for a permutation representation of the group of moves of Rubik’s $2 \times 2 \times 2$ cube, on $21$ points.
(Download the file and call include("rubik2.jl")
in your OSCAR session.)
Study the structure of this permutation group, similar to the study of Rubik’s $3 \times 3 \times 3$ cube in https://nbviewer.org/github/oscar-system/OSCARBinder/blob/master/rubik.ipynb:
Factorization for solving
the puzzle.Find a group $G$ of order $2^9$ such that
One possibility is to use the classification of groups of order $2^9$:
GAP.Globals.SmallGroupsInformation( 2^9 ) shows ranks and $p$-classes of the
groups of order $2^9$.
To which ranges can the group $G$ belong?
(Such a group is constructed by hand in Section 3 of a paper by B. Brewster and G. Yeh (J. Algebra 146 (1992), 18-29).)
Let $A_5$ denote the alternating group on five points. Consider the direct product of $n$ copies of $A_5$, for $n = 2, 3, \ldots$.
Show that for small $n$, it is possible to find a generating set for this direct product that consists of two elements.
Background:
The derived subgroup G’ of a group G is defined as the subgroup
that is generated by the
Find a group G with the property that not every element in G’ is a commutator.
For a given positive integer n,
the ring $R$ of residue classes of integers modulo n
can be created with R, epi = residue_ring(ZZ, n).
For different values of n, compute
The group of invertible $k \times k$ matrices over $R$ can be created as GL$(k, R)$. Compute the order of this group, for various values of $n$, for example increasing powers of a prime.
Let l be a list of pairs,
for example
l = [[1, 1], [2, 1], [1, 2], [2, 2], [2, 3], [3, 1]],
and suppose that the task is to compute the sublist of those pairs
that are the first ones with given first entry.
In the above example, the result would be
[[1, 1], [2, 1], [3, 1]].
An elegant way to solve this task would be
filter(x -> x == l[findfirst(y -> y[1] == x[1], l)], l).
Is this approach a good idea?
Compare this approach with a solution via a for loop,
for different lengths of l.
Consider the cube in Exercise O4, embedded into the 3-dimensional space such that all coordinates of the vertices have absolute value 1.
Then the symmetries can be represented by integral matrices.
Write down the matrices corresponding to the above permutations.
Show that mapping the permutations to the matrices
really defines a group isomorphism.
For that, use the function hom and prescribe images of
the permutation generators,
or via the action of the matrix group on the orbit of the vector
[1, 1, 1].
Construct the various actions mentioned in Exercise O4 (on vertices, edges, faces, etc.) as actions of the matrix group on orbits of suitable sets of row vectors.