192 Normat 58:4, 192 (2010)

Summary in English

Ulf Persson, Oktahedergruppen och

dess generaliseringar III - Oktaplexen.

(Swedish).

The octaplex is a self-dual regular po-

lyhedron in 4-dimensional space, con-

sisting of 24 octahedral cells. It has

no analogue in any other dimension. It

can easily be constructed out of the 4-

dimensional hypercube (8-cell), by eit-

her adding vertices corresponding to

each of the eight cubical cells, or consi-

dering mid-points of its 24 edges. There

is a rich combinatorial structure. The

vertices split up naturally in three di-

sjoint sets each making up a 16-cell (the

dual to the 8-cell, consisting of tetrahed-

ronal cells), any two of which make up

a 8-cell. Those sub-polyhedra are per-

muted under the symmetry group of the

Octaplex which turn out to be a non-

normal extension of the 4-dimensional

octahedral group of index three.

The symmetry group of the Octap-

lex is then considered in some detail,

identifying all of its conjugacy classes

and their geometric interpretations. Dif-

ferent projections of the Octaplex are

presented, some of which show startling

symmetries, as can be gleaned from the

cover.

If R

4

is naturally identiﬁed with the

Hamiltonian quaternions H one of the

16-cells can be identiﬁed with the qua-

ternion subgroup Q of H

∗

given by

±1, ±i, ±j, ±k, while more spectacular-

ly the full octaplex O of 24 elements,

also make up a subgroup of H

∗

. Note

that O is not isomorphic with S

4

. The

symmetry group of the octaplex O can

be represented as given by linear trans-

formations of the type x 7→ axb and a

complete identiﬁcation of the elements

a, b ∈ H occurring is given. Those make

up two distinct Octaplexes O, O

0

, dual

to each other.

Finally the solvable structure of the

group is presented in a variety of diﬀe-

rent ways.

Ulf Persson, Appendix: Heisenberg-

gruppen (Swedish)

This is a proof of the claim that a na-

turally appearing subgroup of 32 ele-

ments in the previous article (namely

Q × Q divided out by the subgroup

(1, 1), (−1, −1) where Q is the quater-

nion group of 8 elements) is actually

isomorphic with the discrete Heisenberg

group H

2

(Z/2Z) of 32 elements of 4 × 4

matrices of type





1 a c

0 I

2

b

0 0 1





where a, b ∈ Z

2

2

, c ∈ Z

2

and I

2

denotes

a 2 × 2 identity matrix.

Marc Bezem, Euclid’s Lemma Revi-

sited(English).

A ’minimal’ proof of the fact that a pri-

me p dividing ab divides at least one of

the factors a, b is presented without in-

voking the Euclidean algorithm.