Exercises for OSCAR, part 2

O8^*. Generators for direct products of groups

Let A₅ denote the alternating group on five points. Consider the direct product of n copies of A₅, for n = 2, 3, .... Show that for small n, it is possible to find a generating set for this direct product that consists of two elements.

(Hint: Try the function rand.)

(Background:

• A generating set of cardinality 2 exists if and only if n is at most 19, see P. Hall's paper "The Eulerian functions of a group" (1935).
• This phenomenon does not occur for groups of prime power order.)

O9^*. Non-commutators in the derived subgroup

The derived subgroup G' of a group G is defined as the subgroup that is generated by the commutators x^-1 y^-1 x y of elements x, y in G.

Find a group G with the property that not every element in G' is a commutator.

(A brute force approach can be implemented as follows. For a given small n, all_small_groups( n ) returns representatives of isomorphism classes of the groups of order n. For these groups, compute the derived subgroup and the commutators.)

O10^*. Study residue class rings

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 number of solutions of x² = -1 in R,
• the number of units (i. e., invertible elements) in R,
• the number of idempotents in R (i. e., elements x with the property x² = x).

The group of invertible k by 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.

O11^*. Bad programming

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.

O12^*. Symmetries of a cube (as matrices)

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 vectors.