Math 60210, Basic Algebra, Problem Set 6, Fall 2009
due Tues, October 13
Do 6 of these problems; note that some of the problems count as more than one problem. For
example, if you do 10-11-12, you only need to do three other problems.
1. Let Gbe a finite group with normal subgroup N. Prove that there exists a composition
series G0={e} ⊂ G1⊂ · · · ⊂ Gr=Gsuch that Gi=Nfor some i.
2. Let G=Z/28Z. Find two different composition series for Gand verify that the composition
factors are the same up to reordering.
3. (A) Let G=D56. Find a composition series for G.
(B) Explain how to find a composition series for D2nfor any integer n(this can be done in
rough terms, i.e., just give the idea).
4. Let F=Z/3Z. Give a composition series for GL(2, F ).
5. Let Gbe a nonabelian group of order p3where pis prime. Let Z=Z(G), and let C=Z/pZ.
(A) Prove that Z∼
=Cand G/Z ∼
=C×C.
(B) Let Hbe a subgroup of Gof order p2. Prove that His normal and Z⊂H.
6-7. Let Gbe a nonabelian group of order 8. Prove that either G∼
=D8or G∼
=Q8(there are
hints on the course website).
8. Let Fbe a field and assume |F|>2. Show that T(n, F ) is solvable but not nilpotent, and
show that U(n, F ) is nilpotent.
9. Let Gbe a finite group with normal subgroup H. Let pbe a prime dividing |H|and suppose
that pand [G:H] are relatively prime. Prove that every p-Sylow subgroup of Gis contained in
H.
10-11-12. Let Gbe a group of order 60 and let n5be the number of 5-Sylow subgroups. Prove
that if n5>1, then Gis a simple group. Prove that A5is simple (you can use the following
series of hints):
(A) Show that n5= 6.
(B) Let Pbe a 5-Sylow subgroup. Show that |NG(P)|= 10 (use proof of Sylow (2) and (3)).
(C) Let H⊂Gbe a normal subgroup and suppose 5 divides |H|. Prove that all six 5-Sylow
subgroups of Gare contained in H.
(D) Let Hbe as in (C). Prove that if His a proper normal subgroup, then |H|= 30.
(E) Let Hbe as in (D). Prove that the number of 3-Sylow subgroups or the number of 5-Sylow
subgroups of Gis one, and conclude that |H|cannot be a multiple of 5.
(F) Suppose there exists a proper normal subgroup Hand |H|= 6. Prove that Ghas a normal
3-Sylow subgroup, which contradicts the assumption that Gis simple.
(G) Suppose there exists a proper normal subgroup of order 12. Prove that Ghas a normal
2-Sylow subgroup or a normal 3-Sylow subgroup.
(H) Suppose there exists a proper normal subgroup Hof order 2, 3 or 4. Prove that G/H has a
normal 5-Sylow subgroup, and use this to prove that Ghas a proper normal subgroup of order
divisible by 5, contradicting (E). Use the correspondence theorem for this, and for one of these
cases, you may want to recall that a group of order 30 has a unique 5-Sylow subgroup.
13.
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