If...... then (1) holds but not (2).
All but a finite number of the Gs are empty.
This will be proved by showing that H has but a single orbit on M.
Go to List of words starting with:
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Y
Z