HW due Fri Nov 10 Mon Nov 13

...since Nov 10 is a holiday.

1. Fix n>0 a natural number.
Call a function f:Z->Z is of period n if f(x+n) = f(x) + n for all x.
Let Jug_n denote the set of such f that are bijections.

A. Show that Jug_n is a group under function composition, i.e. the inverse of a function of period n is again of period n.
B. Show that Jug_n has an easy group homomorphism onto S_n, and that the kernel is isomorphic to Zⁿ.
C. Show that the map Jug_n ->> S_n has a one-sided inverse S_n -> Jug_n (a group homomorphism).
D. Infer that Jug_n is a semidirect product.

2. Let S_n act diagonally on {1,...,n}⁴, i.e. on 4-tuples.
How many orbits are there, as a function of n = 0,1,2,... ?

3. Say G,H are abelian, and G |X H is a semidirect product (i.e. H is the normal one). Assume that G is also normal. Show that G |X H is abelian.

4. Let Ad g:G -> G denote the function h |-> g h g^-1.
A. Show that Ad g is an automorphism of G.
B. Show that the map Ad: G -> Aut(G) is a group homomorphism.
C. What is its kernel?
D. Show that its image is a normal subgroup of Aut(G).

[H]
1.5 #6,11,16