## Algebra Prelim September 10, 2012

Do as many of the eight problems as you can.
Four completely correct solutions will be a pass; a few complete solutions will count more than many partial solutions.
Always carefully justify your answers.
If you skip a step or omit some details in a proof, point out the gap and, if possible, indicate what would be required to fill it in.

1. Classify all groups of order 2012 up to isomorphism.
   (Hint: 503 is prime.)

2. For any positive integer n, let $G _ { n }$ be the group generated by a and b subject to the following three relations:

$$
a ^ { 2 } = 1 , \quad b ^ { 2 } = 1 , \quad { \mathrm { a n d } } \quad ( a b ) ^ { n } = 1 .
$$

(a) Find the order of the group $G _ { n }$

(b) Classify all irreducible complex representations of $G _ { 4 }$ up to isomorphism.

3. Let R be a (commutative) principal ideal domain, let M and N be finitely generated free R-modules, and let $\varphi : M \to N$ be an R-module homomorphism.

(a) Let K be the kernel of ϕ. Prove that K is a direct summand of M .

(b) Let C be the image of $\varphi .$ . Show by example (specifying R, M, N and $\varphi )$ that C need not be a direct summand of N.

4. Let G be an abelian group.
   Prove that the group ring Z[G] is noetherian if and only if G is finitely generated.

5. Let $M _ { 3 } ( \mathbb { R } )$ be the 3 × 3-matrix algebra over the real numbers R. For any $b \in \mathbb { R }$ , let $B \in M _ { 3 } ( \mathbb { R } )$ be the matrix $\left( \begin{array} { c c c } { { 1 } } & { { b } } & { { 0 } } \\ { { b } } & { { 1 } } & { { b } } \\ { { 0 } } & { { b } } & { { 1 } } \end{array} \right)$ . Find the set of numbers b so that the matrix equation $X ^ { 2 } = B$ has at least one, and only finitely many, solutions in $M _ { 3 } ( \mathbb { R } )$

6. Determine the Galois groups of the following polynomials over Q.

$$
( \mathrm { a } ) \ f ( x ) = x ^ { 4 } + 4 x ^ { 2 } + 1
$$

$$
( \mathrm { b } ) \ f ( x ) = x ^ { 4 } + 4 x ^ { 2 } - 5
$$

7. Prove that if A is a finite abelian group, then Hom ${ \mathrm { \Omega } } _ { \zeta } ( A , \mathbb { Q } / \mathbb { Z } ) \cong \operatorname { E x t } _ { \mathbb { Z } } ^ { 1 } ( A , \mathbb { Z } ) \cong A$ . (Here, $\mathrm { E x t } _ { \mathbb { Z } } ^ { 1 } ( - , - )$ is also sometimes written as $\mathrm { E x t } ( - , - ) . ,$ )

8. Let A be the C-algebra $\mathbb { C } [ x , y ]$ , and define algebra automorphisms σ and τ of A by

$$
\sigma ( x ) = y , \quad \sigma ( y ) = x
$$

and

$$
\tau ( x ) = x , \quad \tau ( y ) = \xi y ,
$$

where $\xi \in \mathbb { C }$ is a primitive third root of unity (namely, $\xi \neq 1$ and $\xi ^ { 3 } = 1 )$ ). Let G be the group of algebra automorphisms of A generated by σ and τ . Define

$$
A ^ { G } = \{ f \in A \mid \phi ( f ) = f { \mathrm { ~ f o r ~ a l l ~ } } \phi \in G \} .
$$

Then $A ^ { G }$ is a subalgebra of $A - \mathrm { y o u }$ do not need to prove this.
Describe the algebra $A ^ { G }$ by finding a set of generators and a set of relations.