   Exercises

### Exercises

E: Consider a G-set X, a normal subgroup U lefttriangleeq G, and the corresponding restriction UX. Check the following facts:
• For each orbit U(x) and any g ÎG, the set gU(x) is also an orbit of U on X.
• The orbits of U on X form a G/U-set, in a natural way.
• The orbits of G/U on U \\X are just the orbits of G on X.
• The U-orbits which belong to the same G-orbit are of the same order.
E: Prove that the number of Sylow p-subgroups divides the order of G and is congruent 1 modulo p, for each prime divisor of the order of G.
E: Prove the statements of the example.
E: Show that E bar (G) is normal in bar (H) bar (G) and in [ bar (H)] bar (G), and that bar (H) bar (G) is not in general normal in [ bar (H)] bar (G). Check that the factor group bar (H) bar (G)/E bar (G) is isomorphic to bar (H), while [ bar (H)] bar (G)/[ bar (H)] bar (E) is isomorphic to bar (G). What does this mean, in the light of exercise, for the enumeration of the orbits of bar (H) bar (G) and [ bar (H)] bar (G)?
E: Check the lemma above.

harald.fripertinger@kfunigraz.ac.at,
last changed: August 28, 2001   Exercises