Math 661
Homework 2 Solutions
Fall 2013
Drew Armstrong
1. Consider the lattice of subgroups L (G) of a group G. For each H L (G) and g G let
gHg 1 := cfw_ghg 1 : h H.
(a) Show that gHg 1 is a subgroup of G.
(b) Show that the map G L (G) L (G) dened by (g,
Math 661
Homework 5
Fall 2013
Drew Armstrong
Problem 1 (Splitting Lemma). Let R be a commutative ring with 1 and consider a short
exact sequence of R-modules:
q
r
0 A B C 0.
Prove that if there exists t : B A such that t q is the identity on A, then B A C
Math 661
Homework 1 (Review of 561/562)
Fall 2013
Drew Armstrong
1. Let H G be a subgroup. Call the identity element 1.
(a) State the denition of equivalence relation.
(b) Dene a relation on G by setting a H b a1 b H. Prove that this is an
equivalence rel
Math 661
Homework 2
Fall 2013
Drew Armstrong
1. Consider the lattice of subgroups L (G) of a group G. For each H L (G) and g G let
gHg 1 := cfw_ghg 1 : h H.
(a) Show that gHg 1 is a subgroup of G.
(b) Show that the map G L (G) L (G) dened by (g, H) gHg 1
Math 661
Homework 3
Fall 2013
Drew Armstrong
1. Given a group, dene its center (Zentrum):
Z(G) := cfw_g G : gh = hg for all h G.
Note that Z(G) is abelian and Z(G)
2. Let
(a)
(b)
(c)
G. If G/Z(G) is cyclic, show that G is abelian.
p be prime and consider
Math 661
Final Exam
Fall 2013
Drew Armstrong
1. Let G be a group and consider the group homomorphism : G Aut(G) which sends
g G to the map x gxg 1 in Aut(G). The orbits Orb(x) := cfw_gxg 1 : g G are called
conjgacy classes and the stabilizers C(x) := cfw_
Math 661
Homework 3
Fall 2013
Drew Armstrong
1. Given a group, dene its center (Zentrum):
Z(G) := cfw_g G : gh = hg for all h G.
Note that Z(G) is abelian and Z(G)
G. If G/Z(G) is cyclic, show that G is abelian.
Proof. Assume that G/Z(G) is cyclic. Then w
Math 661
Homework 4
Fall 2013
Drew Armstrong
1. Let K be a eld and let V be a K-module. A composition series of length n is a chain of
submodules
0 = V0 < V1 < < Vn = V
such that each quotient Vi+1 /Vi is a simple K-module (has no nontrivial submodules).
Math 661
Homework 5
Fall 2013
Drew Armstrong
Problem 0 (Abelianization). Let G be a group and for all g, h G dene the commutator [g, h] := ghg 1 h1 G. The subgroup of G generated by commutators is called the
commutator subgroup:
[G, G] := [g, h] : g, h G
Math 661
Homework 1 Solutions
Fall 2013
Drew Armstrong
1. Let H G be a subgroup. Call the identity element 1.
(a) State the denition of equivalence relation.
Proof. Let R G G be a subset and write a b to mean that (a, b) R. We
say that is an equivalence r