Hints for Homework 13
9.57: The hint Gallian has in the back
is a good one, but you need Exercise 39. Let me present that
as a lemma.
Lemma: Let H be a normal subgroup of a finite group G. Then
the order of an element gH of G/H divides the order of g in G.
Proof: Let n be the order of g. Then (gH)^n = (g^n)H = H
(since g^n = e). Recall that H is the identity of G/H.
Therefore, the order of gH divides n, which is the order of g.
Now let gH be an element of order n.
Use the lemma to say something about the order of g,
then find a power of g that solves the problem.
Please include the lemma and rewrite or recopy the proof
in your solution to this exercise.
Don't forget the second part of the exercise which asks
you to show that G must be finite to have this divisibility
property.
For 9.18, write down the cosets of U(20)/U_5(20) (you should
find four of them) and write down the 4-by-4 multiplication
table for the cosets. Remember that (aH)(bH) = abH.
For 9.31, second part, use the theorem that the internal direct
product is isomorphic to the external direct product.
For 9.58, use inner automorphisms and the Normal Subgroup Test.
For the four exercises in Chapter 10, I don't think there is
anything particularly tricky -- just show that the group
operation is preserved. We discussed the determinant and the
derivative maps in class.