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.

Exercises