Exercises for OSCAR

O1. Some large numbers

Let p₁, p₂, p₃, ... be the sequence of prime integers, and set E_n = p₁ p₂ ... p_n + 1.

Compute the values E_n for n = 1, 2, 3, ...; which of them are prime integers?

O2. Subgroups of the symmetric group

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).

O3. An isomorphism

Show that the groups SL(2,4) and A₅ are isomorphic.

(Hint: There is an OSCAR function isomorphism. Alternatively, consider the action of SL(2,4) on the right cosets of the normalizer of a Sylow 2-subgroup, see sylow_subgroup, normalizer, right_transversal.)

O4. Symmetries of a cube (as permutations)

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?

O5. Some finite triangle groups

For n = 2, 3, 4, 5, compute a faithful permutation representation of the group < x, y; x² = y³ = (x y)ⁿ = 1 >, determine the group orders; try also describe.

(Try isomorphism. For larger values of n, these groups are infinite.)

O6. Rubik's 2x2x2 cube

The file rubik2.jl contains Oscar input for a permutation representation of the group of moves of Rubik's 2x2x2 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 3x3x3 cube in https://nbviewer.org/github/oscar-system/OSCARBinder/blob/master/rubik.ipynb.

(What is the order of the group? Construct the action homomorphism for the action on a system of blocks: What is the image, what is the kernel? Use the GAP library function Factorization for solving the puzzle.)

O7. Find a group with given properties

Find a group G of order 2⁹ such that Z(G) has order 2, G' is elementary abelian of order 2⁵, G/G' is elementary abelian of order 2⁴, Z(G) is contained in G', and G has an automorphism of order 5.

One possibility is to use the classification of groups of order 2⁹: GAP.Globals.SmallGroupsInformation( 2^9 ) shows ranks and p-classes of the groups of order 2⁹. To which ranges can the group G belong?

(Hint: When running over a list of candidates, it is not always advisable to run from the beginning to the end.)

(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).)