Algebra Prelim Fall 1994   Answer All Problems

1.
Let p be a prime, let G be a finite p-group, let Z be the center of G, and let 1$ \ne$H $ \vartriangleleft$ G.

(i) Let x $ \in$ H$ \backslash$Z, and let $ \mathfrak {C} $(x) denote the conjugacy class containing x. Prove that $ \mathfrak {C} $(x) $ \subseteq$ H and p divides |$ \mathfrak {C} $(x)|.

(ii) Prove that Z $ \cap$ H$ \ne$1.

(iii) Let A be a maximal normal abelian subgroup of G. Prove that A is also a maximal abelian subgroup of G. (Apply (ii) with G = G/A and H the centralizer of A in G.)

2.
Let G be a simple group of order 180.

(i) Prove that the number of 5-Sylow subgroups of G is 36.

(ii) Prove that the normalizer of a 3-Sylow subgroup of G has order 18.

(iii) Prove that the 3-Sylow subgroup of a group of order 18 is normal in that group.

(iv) If A and B are distinct 3-Sylow subgroups of G, prove that A $ \cap$ B = 1 (consider the centralizer in G of A $ \cap$ B).

(v) Prove that there is no simple group of order 180.

3.
Let R be a PID (principal ideal domain), and let M be a cyclic left R-module. Suppose M = A $ \oplus$ B where A and B are nonzero left R-modules. Prove that there exists r $ \in$ R$ \backslash$ 0 such that rM = 0. Prove further that for such an r, there exist distinct primes p, q $ \in$ R such that pq divides r.

4.
Let R be the ring $ \mathbb {Z} _{(2)}^{}$[[X]]/(X - 2), where $ \mathbb {Z} _{(2)}^{}$[[X]] denotes the power series ring in X over $ \mathbb {Z} _{(2)}^{}$ , the localization of $ \mathbb {Z} $ at the prime 2.

(i) If q is an odd integer, prove that q is invertible in $ \mathbb {Z} $[[X]]/(X - 2).

(ii) Define $ \theta$ : $ \mathbb {Z} $[[X]]$ \to$$ \mathbb {Z} _{(2)}^{}$[[X]] by $ \theta$$ \sum$aiXi = $ \sum$aiXi, and let $ \pi$ : $ \mathbb {Z} _{(2)}^{}$[[X]]$ \to$R be the natural epimorphism. Prove that $ \pi$$ \theta$ is surjective and deduce that R $ \cong$ $ \mathbb {Z} $[[X]]/(X - 2).

(iii) Prove that R $ \ncong$ $ \mathbb {Z} $[X]/(X - 2).

5.
Let p be a prime, let K $ \subseteq$ L be fields of characteristic p, let $ \alpha$,$ \beta$ $ \in$ L, and let d be a positive integer. Suppose [K($ \alpha$) : K] = d, [K($ \beta$) : K] = p, $ \alpha$ is separable over K, and $ \beta$ is not separable over K.

(i) Prove that K($ \alpha$) = K($ \alpha^{p}_{}$) and $ \beta^{p}_{}$ $ \in$ K.

(ii) Prove that K($ \alpha$) $ \subseteq$ K($ \alpha$ + $ \beta$).

(iii) Prove that [K($ \alpha$ + $ \beta$) : K] = pd.

6.
Let p be a prime, and let K = $ \mathbb {C} $(t), the quotient field (field of fractions) of the polynomial ring $ \mathbb {C} $[t].

(i) Prove that Xp - t is irreducible in K[X].

(ii) Let L be the splitting field of Xp - t over K. Determine the Galois group of L over K.


 
Peter Linnell
1998-06-07