Exercises |

E:Consider aG-setX, a normal subgroupU lefttriangleeq G, and the corresponding restriction. Check the following facts:_{U}X

- For each orbit
U(x)and anyg ÎG, the setgU(x)is also an orbit ofUonX.- The orbits of
UonXform aG/U-set, in a natural way.- The orbits of
G/UonU \\Xare just the orbits ofGonX.- The
U-orbits which belong to the sameG-orbit are of the same order.

E:Prove that the number of Sylowp-subgroups divides the order ofGand is congruent 1 modulop,for each prime divisor of the order ofG.

E:Prove the statements of the example.

E:Show thatEis normal in^{ bar (G)}bar (H)and in^{ bar (G)}[ bar (H)], and that^{ bar (G)}bar (H)is not in general normal in^{ bar (G)}[ bar (H)]. Check that the factor group^{ bar (G)}bar (H)is isomorphic to^{ bar (G)}/E^{ bar (G)}bar (H), while[ bar (H)]is isomorphic to^{ bar (G)}/[ bar (H)]^{ bar (E)}bar (G). What does this mean, in the light of exercise, for the enumeration of the orbits ofbar (H)and^{ bar (G)}[ bar (H)]?^{ bar (G)}

E:Check the lemma above.

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

Exercises |