Sufficiently large implies character-separating
From Groupprops
This fact is related to: linear representation theory
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Statement
Let be a finite group and
a field whose characteristic does not divide the order of
. Then, if
is a sufficiently large field for
(that is,
contains all the
roots of unity where
is the exponent of
), then
is a character-separating field for
.
By is character-separating for
, we mean that given two distinct conjugacy classes
and
of
and elements
, there exists a linear representation
whose character takes different values on
and
.
Facts used
- Sufficiently large implies splitting: If
is a sufficiently large field for
, then
is a splitting field for
: every representation of
over
is completely reducible, and every representation irreducible over
is irreducible over any field extension of
.
- Splitting implies character-separating: Any splitting field for a finite group is character-separating: given any two conjugacy classes, there is a linear representation whose character takes different values on these conjugacy classes.