Invariants for finite groups

Constructing Invariants for Finite Groups

on some invariants of finite groups

in this note we are going to survey several invariants of finite groups related either to their orders or to generating sets or to lattices of subgroups. some relations among these invariants will be exhibited. special attention will be paid to monotonicity of them.

Finite Simple Groups and Dickson Invariants

In a recent paper [1], Adem, Maginnis and Milgram calculated the mod 2 cohomology of the Mathieu simple group M12. An interesting feature of the answer they obtain is that it is Cohen–Macaulay. There is a subalgebra isomorphic to the rank three Dickson invariants, namely a polynomial ring in generators of degrees 4, 6 and 7, and over this subalgebra the whole cohomology ring is a finitely gener...

Rings of Invariants of Finite Groups

We shall prove the fundamental results of Hilbert and Noether for invariant subrings of finite subgroups of the general linear groups in the non-modular case, i.e. when the field has characteristic zero or coprime with the order of the group. We will also derive the Molien’s formula for the Hilbert series of the ring of invariants. We will show, through examples, that the Molien’s formula helps...

Separating Invariants and Finite Reflection Groups

Roughly speaking, a separating algebra is a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a more geometric notion of separating algebra. This allows us to prove that when there is a polynomial separating algebra, the group is generated by reflections, and when there is a complete int...