Abstract Algebra Answers

If a finite group G has at most three irreducible complex representations, show that G ∼ {1},Z2,Z3 or S3.

Show that any quasi-Frobenius ring is left (and right) Kasch.

identify a basis and write down the dimension for each of the vector spaces.
(i) the line in R^3 given in paramaterized fprm by x=2t, y=-3t, z=t(root5)
(ii)the solution set for a linear system which is given by the general solution
x1 = 2t -3s + r
x2 = t + s - 5r
x3 = -s - r
x4 = t
x5 = s
x6 = r

Show that for any prime p, a group G of order p2 must be abelian.

For finite abelian groups G and H, show that RG ∼ RH as R-algebras iff |G| = |H| and |G/G2| = |H/H2|.

Let G = [x] be a cyclic group of order 5. Show that U(ZG) = [u] × (±G) ∼ Z ⊕ Z2 ⊕ Z5.

Let k be a field of algebraic numbers, and A be its ring of algebraic integers. Let G be any finite group. Using the character of the regular representation of G, give proof for the fact that AG has no idempotents except 0 and 1.

Let k be any field of characteristic 2, and let G = S4. Let M be the kG-module given by
ke1 ⊕• • •⊕ke4/k(e1 + • • • + e4),
on which G acts by permuting the ei’s. Compute the kG-composition factors of M.

Let k be the algebraic closure of Fp and K = k(t), where t is an indeterminate. Let G be an elementary p-group of order p2 generated by a, b. Show that
a → A =
1 1
0 1

b→ B =
1 t
0 1
defines a representation of G over K which is not equivalent to any representation of G over k.