the generators for ~(3)=G98. The orbit of 8 under G (2) =G 9 is { 8,10,11,13,14,16,21 }, and so we calculate that' I A (3) I = 7. Initially, A= {1}, F= {2..7}, andT= {$3,S5,$6}. Choosing T = 2 from F gives g I = s7 = (1,2,3,4,6,5,7). Then 8 gl-t = 8, sog2 is the identity, and we add s7 to T. Thus, A = {1..7}, F= empty, and T= {s3,ss,s6, sT}. Since A has the correct size, we are finished. The new strong generating set is th
Consider now the interchange of 1 and 8, so that the new base will be
[9, 8, 1, 10, 2, 12].
We require the generators for ~(3)=G98. The orbit of 8 under G (2) =G 9 is
{ 8,10,11,13,14,16,21 }, and so we calculate that' I A (3) I = 7. Initially,
A= {1}, F= {2..7}, andT= {$3,S5,$6}.
Choosing T = 2 from F gives
g I = s7 = (1,2,3,4,6,5,7).
Then 8 gl-t = 8, sog2 is the identity, and we add s7 to T. Thus,
A = {1..7}, F= empty, and T= {s3,ss,s6, sT}.
Since A has the correct size, we are finished.
The new strong generating set is the same as the old. By checking not only S (/+2) but also
S (/) for elements that fix 13j+1, we could have saved ourselves the trouble of duplicating
permutations already in S. This simple improvement is used in implementations, We leave it
as an exercise for the reader to make the necessary modifications to
Analysis of Interchanging Base Points
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