Any linear group is finite-dominating in the corresponding affine group over characteristic zero

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Statement

Suppose k is a field of characteristic zero and G is a subgroup of the general linear group GL(n,k). Consider the group A = k^n \rtimes G, i.e., the affine group correpsonding to G. Then, G is a Finite-dominating subgroup (?) in A: in other words, any finite subgroup of A is conjugate in A to a subgroup of G.

Related facts

Applications

Facts used

  1. General linear group is finite-dominating in general affine group over characteristic zero
  2. Intersection of finite-dominating subgroup with any subgroup whose product with it is the whole group is finite-dominating in it

Proof

Given: A group G \le GL(n,k). A is the semidirect product of k^n and G. All the groups G, A, GL(n,k) and GA(n,k) are viewed here as subgroups of GA(n,k).

  1. By fact (1), GL(n,k) is finite-dominating in GA(n,k).
  2. Since A contains k^n, the product of A and GL(n,k) is GA(n,k). Also, the intersection of GL(n,k) with the group A is G. Thus, by fact (2), G is finite-dominating in A.