4. Bijective functions

We first introduce the computer algebra system GAP and then give several examples.

#### 4.1. GAP: Graphs, algorithms, programming

#### 4.1.1. GAP and the alternating group A2<sup>n</sup>

GAP [19] is a system for computational discrete algebra, in particular computational group theory. We use GAP to decide, whether certain fair or unfair update rules generate the full symmetric or alternating group S2<sup>n</sup> or A2<sup>n</sup> , respectively.

Our results so far:

#### Theorem 1.


iii. The ðunfairÞ elementary update rules with exactly one active cell for ECA-57 generate the full alternating group A2<sup>n</sup> for n ¼ 4,…, 28.
