Assigned	Problems
12/9/10	Show that k[G×H] is isomorphic to k[G]⊗k[H] as a k-algebra for all groups G, H. (The tensor product of algebras is always an algebra.)
12/6/10	Set R=k[S3]. If 1/6∈k, show that R is semisimple. Identify R/J(R) if k has characteristic 2 or 3.
12/2/10	Show that the sum of two nilpotent ideals is nilpotent
11/29/10	Show that every complex hermitian matrix (resp. real symmetric matrix) has a basis of eigenvectors.
11/14/10	BA I, 6.1 #5 (the vector space of bilinear forms)
11/11/10	BA I, 3.10 #2,10 (|coker(A)|=det(A); A²=A)
11/8/10	BA I, 3.8 #1 (Abelian group with 3-generators, 2 relations)
11/4/10	BA I, 3.7 #1-2 (Normal form of matrices over Z and Q[λ])
11/1/10	The left ideals of dimension n in Mn(k) are in 1-1 correspondence with points of Pn-1
10/28	BA-I 2.16 #1,3 (Z is integrally closed, and dimQQ(ζp)=p-1.)
10/21	In-class Exam (group theory)
10/14	Let H be the ring of holomorphic functions, S={f | f(0)≠0}. Show that S-1H is the ring of meromorphic functions which are analytic at 0, and that this ring is noetherian.
10/11	(1) If F is a field, show that Mn(F) has no nontrivial ideals. (2) Let I denote the kernel of the ring map Mn(Z/p²) → Mn(Z/p) induced by Z/p² → Z/p. Show that I²=0, i.e., show that AB=0 for every A,B in I.
10/7	Show that every complex representation of Cn is a direct sum of eigenspaces Vk={v∈V|σv=exp(2πik/n)v} (0≤k<n). Conclude that every irreducible representation of Cn is 1-dimensional.
10/4	If n>3 and G=GLn(F), show that [G,G]=SLn(F). Is SLn(Fp) a simple group for n even? (p prime>2).
9/30	(1) If every Sylow subgroup Pi in G is normal, show that G=∏ Pi.   (Cf. BAI, 4.6 #9,11) (2) If G is a finite nilpotent group, show that every Sylow subgroup is normal, and hence G=∏ Pi.
9/27	If |G|=12, show that G has a normal Sylow subgroup. Using this, show that there are only 5 isomorphism classes of groups of order 12, including D6, A4, and a metacyclic group.
9/23	Show that every group of order 6 is isomorphic to either C6 or S3;   (Cf. BAI, 1.13 #3,4) If q<p and |G|=pq, show that G has a normal Sylow p-subgroup
9/20	BAI, 1.12 #10;    Show that the group of monomial matrices in GLn(F) is isomorphic to the wreath product G ι Sn
9/16	(1) Show than Dn=<r,t | r^n,t^2,rtrt>;    (2) If f:G→G' is onto, and H→G' is any group map,         show that the kernel of G×G'H→ H is isomorphic to ker(f)
9/13	BAI, 1.5 #5;  1.6 #2;  1.8 #11;  1.9 #1
9/2	BAI, 1.2 #5,13; Classify all groups of orders 2 and 3 Show that GL2(F2) and D3 are isomorphic

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