Download Algebra Autumn 2013 Frank Sottile 24 October 2013 Eighth Homework

yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Birkhoff's representation theorem wikipedia, lookup

Ring (mathematics) wikipedia, lookup

Tensor product of modules wikipedia, lookup

Polynomial ring wikipedia, lookup

Commutative ring wikipedia, lookup

Algebra Autumn 2013
Frank Sottile
24 October 2013
Eighth Homework
Hand in to Frank Tuesday 29 October:
44. [20] The wreath product Sm ≀ Sn of symmetric groups is the semidirect product (Sm )n ⋊ϕ Sn where
ϕ is the action of Sn on (Sm )n permuting the factors of (Sm )n .
(a) For (π1 , . . . , πn , ω) ∈ Sm ≀ Sn (πi ∈ Sn and ω ∈ Sn ) define the map from [m] × [n] to itself by
(π1 , . . . , πn , ω).(i, j) := (πω(j) (i) , ω(j)) .
(Here, [m] := {1, . . . , m} and the same for [n]. Show that this defines an action of Sm ≀ Sn on
[m] × [n].
(b) Using this action or any other methods show that S2 ≀ S2 ≃ D8 , the dihedral group with 8
(c) This action realizes S3 ≀ S2 as a sugroup of S6 . What are the cycle types of permutations of
S3 ≀ S2 ? For each cycle type, how many elements of S3 ≀ S2 have that cycle type?
Hand in for the grader Tuesday 29 October:
45. Suppose that R is a commutative ring with characteristic p, a prime. Prove that the map F : R → R
defined by F (r) = rp is a ring homomorphism. Hint: You may first need to prove the binomial
theorem for this ring.
46. An element x in a ring R is nilpotent if there is a positive integer n with xn = 0. Prove that the
set η(R) of nilpotent elements of a commutative ring R forms an ideal, called the nilradical of R.
Show that η(R/η(R)) = {0}.