Special linear group is fully characteristic in general linear group
From Groupprops
This article gives the statement, and possibly proof, of a particular subgroup or type of subgroup (namely, Special linear group (?)) satisfying a particular subgroup property (namely, Fully characteristic subgroup (?)) in a particular group or type of group (namely, General linear group (?)).
Statement
Let be a field and
be a natural number. Let
denote the group of invertible
matrices over
under multiplication and
denote the subgroup comprising matrices of determinant one.
is a fully characteristic subgroup of
: every endomorphism of
restricts to an endomorphism of
.
Facts used
- Commutator subgroup of general linear group is special linear group: This is true except in the case where
and the field
has exactly two elements.
- Commutator subgroup is fully characteristic
Proof
For the field with two elements
In this case, every element of has determinant one, so
. Since every group is fully characteristic as a subgroup of itself,
is a fully characteristic subgroup of
.
For other cases
In all other cases, the result follows from facts (1) and (2).