Download Exercise Batch 7 Taken from John B. Fraleigh, A First Course in

Exercise Batch 7
Taken from John B. Fraleigh,
A First Course in Abstract Algebra, Fifth Edition
Exercise Set 1 From Page 170 ff: 1-15, 17-20, 22, 23, 24-33, 35, 36, 40-44.
As we said in class, if G is a group, and if X is a set, then G acts on X if there
is a homomorphism G → SX . The kernel of this homomorphism is usually
called the kernel of the action. If the kernel is trivial, then we say that the
action is faithful.
In each case below, I’ve sketched a group action. You should state carefully
what the set X is, prove that what is described is really a group action, and
compute the kernel of the action. Also try to determine whether the action is
transitive. (In many cases, there won’t be enough information to tell whether
the action is transitive.)
1. G acts on itself by left multiplication.
2. G acts on itself by conjugation, i.e., if g, x ∈ G, then g : x 7→ gxg −1 .
3. Let H ≤ G; G acts on left cosets of H by left multiplication.
4. G acts on the set of its subgroups by conjugation.
5. Let G ≤ Sn , and fix an integer k, 0 ≤ k ≤ n. Then G acts on the nk
subsets of Nn of size k. If G = Sn is the action transitive? What about if
G = An ?
6. Let G ≤ GLn (F), and fix an integer k, 0 ≤ k ≤ n. Then G acts on the
k-dimensional subspaces of the vector space of n × 1 column vectors with
entries in the field F. If G = GLn (F) is the action transitive?
Exercise Set 2 From Page 183 ff: 4-8, 11, 12-14, 18-21, 23.