computer aided intuition in abstract algebra

Two examples are given in which the computer was used to supplement intuition in abstract algebra. In the first example, the computer was used to search Cayley tables of 4 element groupoids to find those which are 5-associative but not 4-associative. (n-associative means that the product of any n elements is independent of the way the factors are grouped by parentheses.) The computer generated examples suggested the existence of n element groupoids which are (2^n?2+1)-associative but not (2^n-2)-associative, for each integer n≧4.In the second example, the computer counted the numbers g₂(m) of invertible 2×2 matrices with entries chosen from the ring Z_mof integers, for m = 2, 3, 4,…, 18. The insight gained from these results led to a proof that there are

invertible n×n matrices over Z_m.Some applications to graduate and undergraduate instruction are indicated.