Homework 3, due 9/11/2015
“The king is dead, long live the king!”
1. Artin 2.4.6
2. Artin 2.4.10
3. Artin 2.4.11(a)
4. Artin 2.5.1
5. Artin 2.6.4, and show that in a group any two conjugate elements have
the same order. Thus, ab and ba always have the same order.
6. Artin 2.8.5
7. (Bonus problem) Recall from class the group homomorphism Sn → GLn (C)
given by associating to each permutation σ ∈ Sn the corresponding permutation matrix pσ . Let B denote the (subgroup of!) upper-triangular
matrices in GLn (C). Show that every element g ∈ GLn (C) can be expressed in the form g = b1 pσ b2 for some b1 , b2 ∈ B and some permutation
σ ∈ Sn . Moreover show that σ (equivalently, pσ ) is uniquely determined
by g (b1 and b2 are not uniquely determined). Symbolically,
G
GLn (C) =
Bpσ B,
σ∈Sn
F
where “ ” denotes disjoint union. You may want to start with the case
n = 2 for intuition; for the general case, think about row operations.
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