Assume X to be a G-set and check carefully that
g -> bar (g) is in fact a permutation representation, i.e. that
bar (g) ÎSX and that
bar (g1 ·g2)= bar (g1) ·bar (g2).
Prove that ~G is in fact an equivalence relation.
Let GX be finite and transitive. Consider an
arbitrary x ÎX and prove that
| Gx \\X | =(1)/( | G | )
åg ÎG | Xg | 2.
Check that the G-isomorphy simeq (and hence also the
G-similarity ») is an equivalence relation on group actions.
Consider the following definition: We call actions GX and GY
if and only if there exists a pair ( h, q) such that
GX simeq GY and where h is an inner automorphism
, which means that
h:G -> G :g -> g'gg'-1,
for a suitable g' ÎG. Show that this equivalence
relation has the same classes as ».
last changed: August 28, 2001