Sufficiently large implies splitting for every subquotient
A Sufficiently large field (?) for a finite group is a Splitting field (?) for every Subquotient (?) of the group. In particular, it is a splitting field for every subgroup as well as for every quotient group.
Statement with symbols
The converse of the statement is true. In fact, if a field is a splitting field for every subgroup of a finite group, then it is sufficiently large. For full proof, refer: Splitting field for every subgroup implies sufficiently large
- Splitting not implies sufficiently large
- Splitting field for a group not implies splitting field for every subgroup
- Splitting field for a group implies splitting field for every quotient
- Sufficiently large implies splitting
- Exponent of subgroup divides exponent of group
- Exponent of quotient group divides exponent of group
Given: A group , a sufficiently large field for . Subgroups with normal in .
To prove: is a splitting field for .
Proof: By facts (2) and (3), the exponent of divides the exponent of . By the definition of sufficiently large, the polynomial splits completely into distinct linear factors over , where is the exponent of . Let be the exponent of . Then the polynomial , being a factor of , also splits completely into linear factors over . (A simpler way of saying this is that a field containing primitive roots of unity also contains primitive roots of unity for ).
Thus, is sufficiently large for . Hence, by fact (1), is a splitting field for .