Extending finite group actions on surfaces over

Let O E g (resp. C E g and A E g ) and resp. O E g o be the maximum order of finite (resp. cyclic and abelian) groups G acting on the closed orientable surfaces Σ g which extend over ( S 3 , Σ g ) among all embeddings Σ g → S 3 and resp. unknotted embeddings Σ g → S 3 . known that O E g o ⩽ 12 ( g − 1 ) , and we show that 12 ( g − 1 ) is reached for an unknotted embedding Σ g → S 3 if and only if g = 2 , 3 , 4 , 5 , 6 , 9 , 11 , 17 , 25 , 97 , 121 , 241 , 601 . Moreover A E g is 2 g + 2 ; and C E g is 2 g + 2 for even g, and 2 g − 2 for odd g. s are made to see intuitively how these maximal symmetries are embedded into the symmetries of the 3-sphere.