Homework 4, due 9/18/2015
1. Artin 2.5.6 (Recall that the center of a group G is the subgroup
Z(G) = {z ∈ G : xz = zx for all x ∈ G}.)
2. Artin 2.6.10
3. Artin 2.8.3
4. Artin 2.8.6
5. Artin 2.11.4
6. Artin 2.12.2
7. Artin 2.12.4
8. (Bonus Problem) Let SL2 (Z) denote the set of 2-by-2 matrices with integer
entries and determinant equal to one. Similarly, for any integer N > 1,
let SL2 (Z/N ) denote the set of 2-by-2 matrices with entries in Z/N and
determinant equal to one (in Z/N ). First check that SL2 (Z) and SL2 (Z/N )
are both groups under multiplication (note that it makes sense to multiply
elements of SL2 (Z/N ) using the usual matrix multiplication rules, since we
can add and multiply elements of Z/N ). Now, there is a natural ‘reduction
modulo N ’ map
fN : SL2 (Z) → SL2 (Z/N ).
Show that fN is a homomorphism, describe the kernel of fN , and prove
that fN is surjective (this last point is the heart of the problem).
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