Central automorphism group of general linear group
Let be a field and be a natural number. The central automorphism group of general linear group, or inner-centralizing automorphism group of general linear group, is defined as the group of automorphisms of the general linear group the form:
where is a homomorphism from the multiplicative group of to itself, such that the map is an automorphism from the multiplicative group of to itself.
We denote this group by .
Note that this is precisely the group of inner-centralizing automorphisms of the general linear group, i.e., the automorphisms that commute with all the inner automorphisms, or equivalently, that induce the identity map on the projective general linear group. It is thus the kernel of the homomorphism . This group also turns out to be the center of the automorphism group of .
The case of a finite field
For a finite field with elements, all such automorphisms are of the form:
where is some positive integer with the property that is relatively prime to . is viewed modulo : two values of yield the same automorphism iff they are congruent modulo .
The composite of the automorphism corresponding to and the automorphism corresponding to is the automorphism corresponding to . The inverse of the automorphism correspond to is the automorphism corresponding to , where the inverse is taken modulo .
The nature and size of this group
When is relatively prime to , we can identify this with the automorphism group of the multiplicative group of , because the permissible s correspond uniquely to the values that are relatively prime to , and the composition of these works like multiplying the corresponding . In particular, this is an abelian group of order , where is the Euler phi-function. In the case that there is a gcd , the abelian group has size where is the largest divisor of that contains only those prime divisors of that are already divisors of .