MATH 400 – Directed Readings in Module and Representation Theory Final Problem Set due: By email or hardcopy, no later than Saturday, May 15

Groundrules: You may consult only our text, Dummit and Foote, Abstract Algebra. If you use a result from a chapter we have not discussed, please “footnote” it.

I. Let K, K′, P, P′, M be modules over a ring R. Assume that P and P′ are projective R-modules and that we have short exact sequences

0 ⟶ K ⟶ P ⟶ M ⟶ 0

and

0 ⟶ K′ ⟶ P′ ⟶ M ⟶ 0

(with the same M in the right-most nonzero position in both). In this problem you will show that K ⊕ P′ ≃ K′ ⊕ P as R-modules.

A. Using the fact that P is projective show that there is a homomorphism w : P → P′ and a

diagram

\xymatrix{0 \ar[r] & K \ar[r]^{i} & P \ar[r] \ar[d]^{w} & M \ar[d]^{id} \ar[r] & 0 \\ 0 \ar[r] & K′ \ar[r]_{j} & P′ \ar[r] & M \ar[r] & 0}

in which the square commutes.

B. Continuing from part A, show that there is a homomorphism u : K → K′ such that the left

square in

\xymatrix{0 \ar[r] & K \ar[r]^{i} \ar[d]^{u} & P \ar[r] \ar[d]^{w} & M \ar[d]^{id} \ar[r] & 0 \\ 0 \ar[r] & K′ \ar[r]_{j} & P′ \ar[r] & M \ar[r] & 0}

is also commutative.

C. Show that the sequence

0 ⟶ K ⟶ P ⊕ K′ ⟶ P′ ⟶ 0

is exact, where the map K ⟶ P ⊕ K′ is x ⟼ (i(x), u(x)) and the map P ⊕ K′ ⟶ P′ is (y, z) ⟼ w(y) − j(z).

D. Deduce that K ⊕ P′ ≃ K′ ⊕ P as R-modules.

II. Let F be a field and let R = M_{n×n}(F).

A. Show that there is a unique irreducible R-module M, up to isomorphism.

B. Let M be the module from part A. Show that Hom_{R}(M, M) is isomorphic to F as a ring.

III. Let F be a field and let R = F [x]/⟨f(x)⟩ for some f(x) ∈ F [x].

A. State and prove a necessary and sufficient condition on f(x) for R to be a semisimple ring. B. Describe the Wedderburn decomposition of R in the case that R is semisimple. (That is,

what are the n_{i} and the Δ_{i} in the isomorphism

R ≃ M_{n_{1}×n_{1}}(Δ_{1}) × ⋯ × M_{n_{k}×n_{k}}(Δ_{k})

given by Wedderburn’s Theorem?)

Read section 19.1, and using the ideas presented there do the following problems: IV. Show that the character table of any finite group is an invertible k × k matrix, where k = the number of conjugacy classes, or the number of irreducible characters. V. Compute the character tables of

A. The group S_{3} × S_{3}.

B. The alternating group A_{5}.