?? 24 semi-simple comm ring structures on {a,b,c,d} ...

?? halving ab/cd ...

12
3ac
4db

ab
c32
d14

?? "halving into ab/cd corresponds to factoring into 12/34" ... ??? ... ?? and vice versa ?? .... ??? this really is that weird thing that i noticed in high school ??? .... ?? still don't quite get all of it though ... the vice versa part .... ???? .... ?? outer automorphism of automorphism group of wreath product here ??? .... ??? _is_ this connected nicely to solution of quartic somehow ??? ....

?? 4! as resulting from making outer inner ??? .... ?? having trouble getting arithmetic to work here .... ???? ....

wreath product here as 8-elt gp ??? .... outer automorphism of order 2, evidently .... ???? .....

?? well, so maybe the semi-direct product here really is some weird 16-elt gp .... ??? ....

?? "element" vs "cross section" of split 2-elt set ....

?? "unordered dual pair of split 4-elt sets" .... ??? ....

?? number of 2+2 decompositions same as number of 2X2 decomposition because of ... outer automorphism here .... ??? .... ????? ....

?? isomorphism between subgroup of 4! preserving a 2+2 decomposition and subgroup preserving a 2X2 decomposition .... ??? .... ???? extending to ..... ???? .... ????? ..... ?? actually, extending to identity automorphism .... ??? ... ???? confusion ... ???? .....

?? is there duality between 6 halves of one 4-elt set and 6 halves of .... ??? .... ??? doesn't seem to quite make sense ??? ...