Stabilizers are Subgroups

Stabilizers are Subgroups

GsG (see Stabilizers).

Proof

  1. 1Gx=x1GGs
  2. (inverses) Suppose gGx, so gx=x. Then:
g1(gx)=g1x(g1g)x=g1xx=g1x

So then g1Gx.
3. (products) Suppose g1,g2Gx. Then:

(g1g2)x=g1(g2x)=g1x=x

So g1,g2Gx.