Benjamin Sambale
Julian Brough
Hand in by: Mo 14. 11. 2016
Group theory
Sheet 3
Exercise 8 (2 + 2 + 2 + 2 + 2 + 2 points). A subgroup H of a group G is called fully invariant in G
if α(H) ⊆ H for every endomorphism α of G.
(a) Show that every fully invariant subgroup of G is characteristic in G.
(b) Show that every subgroup of a cyclic group is fully invariant.
(c) Compute the normal subgroups of S4 . Which of them are characteristic or fully invariant?
Hint: You may use the structure of the conjugacy classes (1st example class).
(d) Show that Z(G) is characteristic in G for any group G.
(e) Show that Z(G) is not always fully invariant in G.
(f) Show that Inn(G) is characteristic in Aut(G) if G is simple.
Hint: Use Exercise 4(c).
Exercise 9 (2 + 2 points). Determine the composition factors and chief factors of S4 and GL(2, F3 ).
Hint: It helps to consider the action of GL(2, F3 ) on the set of 1-dimensional subspaces of F23 .
Exercise 10 (3 points). Show that S6 is generated by two elements but contains a subgroup which is
not generated by two elements.