August 1999 Algebra Prelim Solutions

1. We first factor 480 as 32*3*5. Note that since G is simple, it cannot have a nontrivial subgroup of index $\le$7, because that would mean that G is isomorphic to a subgroup of A₇, which is not possible by Lagrange's theorem.
   1. Let A = P $\cap$ Q and suppose A > 1. Since P, Q $\leqslant$ C_G(A), we see that P < C_G(A). Using Lagrange's theorem, we deduce that | C_G(A)| = 96, 160 or 480. We cannot have 480 because then A would be a central and hence a normal subgroup of G, which would contradict the hypothesis that G is simple. Also we cannot have | C_G(A)| = 96 or 160, because that would mean that G has a subgroup of index 5 or 3. We now have a contradiction, and we conclude that A = 1.

   2. The number of Sylow 2-subgroups is congruent to 1 modulo 2 and divides 15. It cannot be 1 because that would mean that G has a normal Sylow 2-subgroup. Nor can it be 3 or 5, because then G would have a subgroup of index 3 or 5. Therefore G has 15 Sylow 2-subgroups. Next the number of Sylow 3-subgroups is congruent to 1 modulo 3 and divides 96. This number cannot be 1, because that would mean that the Sylow 3-subgroup is normal. Nor can it be 4, because that would yield a subgroup of index 4. Therefore G has at least 10 Sylow 3-subgroups. Finally the number of Sylow 7-subgroups is congruent to 1 modulo 7 and divides 160. This number cannot be 1, because that would mean that the Sylow 7-subgroup is normal. Therefore G has at least 8 Sylow 7-subgroups.

      We now count elements. Since by (a) any two Sylow 2-subgroups intersect trivially, there are 15*31 nontrivial elements whose order is a power of 2. Next any two Sylow 3-subgroups intersect trivially, because a Sylow 3-subgroup has prime order 3, and we see that there are at least 10*2 elements of order 3. Finally any two Sylow 5-subgroups intersect trivially, because a Sylow 5-subgroup has prime order 5, and we deduce that G has at least 6*4 elements of order 5. We now count elements: we find that G has at least 15*31 + 10*2 + 6*4 = 509 nontrivial elements. Since G has only 480 elements altogether, we have now arrived at a contradiction. We conclude that there is no such group G.

2. Since R is a domain and 0$\notin$S, we see that S^-1R is a domain. Also for a, b $\in$ R and s, t $\in$ S, we have a/s = b/t if and only if at = bs. Suppose p/1 divides (a/s)(b/t) in S^-1R. This means that there exists c/u $\in$ S^-1R such that (p/1)(c/u) = (a/s)(b/t), which means that pstc = abu. Since p is prime, we see that p divides at least one of a, b, u. If p divides u, then p/1 is a unit in S^-1R because u/1 is a unit in S^-1R. Therefore we may assume that p does not divide u; without loss of generality, we may assume that p divides a, say pq = a. Then (p/1)(q/s) = a/s and we see that p/1 divides a/s. Therefore if p/1 is not a unit, it is prime and the result follows.

3. Let P be a finitely generated projective k[X]/(X³ + X)-module. Then there is a k[X]/(X³ + X)-module Q and an integer e such that P $\oplus$ Q $\cong$ (k[X]/(X³ + X))^e. Note that X³ + X = X(X + 1)² and k[X]/(X³ + X) $\cong$ k[X]/(X) $\oplus$ k[X]/(X + 1)², so P $\oplus$ Q $\cong$ (k[X]/(X))^e $\oplus$ (k[X]/(X + 1)²)^e. We may view this as an isomorphism of finitely generated k[X]-modules. We use repeatedly without comment that a map between k[X]/(X³ + X)-modules is an isomorphism as k[X]/(X³ + X)-modules if and only if it is an isomorphism as k[X]-modules. Since k is a field, k[X] is a PID, so the structure theorem for finitely generated modules over a PID tells us that
   P $$\cong$$ $$\bigoplus_{i}^{}$$k[X]/(f_i)    and    Q $$\cong$$ $$\bigoplus_{i}^{}$$k[X]/(g_i)
   where the f_i, g_i are either 0 or positive powers of monic irreducible polynomials. Then we have
   $$\bigoplus_{i}^{}$$k[X]/(f_i) $$\oplus$$ $$\bigoplus_{i}^{}$$k[X]/(g_i) $$\cong$$ (k[X]/(X))^e $$\oplus$$ (k[X]/(X + 1)²)^e.
   The uniqueness part of the structure theorem for finitely generated modules over a PID now tells us that f_i = X or (X + 1)² for all i. It follows that a finitely generated k[X]/(X³ + X)-module is isomorphic to a finite direct sum of modules of the form k[X]/(X) or k[X]/(X² + 1).

4. Suppose there is a positive integer n such that MJⁿ = MJ^{n + 1}$\ne$ 0. Then MJⁿ is finitely generated because M is Noetherian, and (MJⁿ)J = MJ^{n + 1} = MJⁿ. By Nakayama's lemma we deduce that MJⁿ = 0, as required.

5. Since R is a right Artinian ring with no nonzero nilpotent ideals, the Wedderburn structure theorem tells us that R $\cong$ R₁ $\oplus$ ... $\oplus$ R_n, where n is a positive integer, and the R_i are matrix rings over division rings. If n > 1, then (1, 0,..., 0) is a nontrivial idempotent, so n = 1 which means that R is a matrix ring over a division ring. If this matrix ring has degree > 1, then the matrix with 1 in the (1, 1) position and zeros elsewhere is a nontrivial idempotent. Therefore R is isomorphic to a matrix ring over a division ring, and the result follows.

6. The character table for S₄ is given below; the irreducible characters are c₁,...,c₅. c₁ is the principal character, c₂ is the character coming from the sign of the permutation, r is the permutation character (not irreducible), c₄ = r - c₁, and c₅ = c₂c₄. The remaining row, the character of c₃, can easily be filled in using the orthogonality relations.

   Class Size	1	6	8	6	3
   Class Rep	(1)	(12)	(123)	(1234)	(12)(34)
   c1	1	1	1	1	1
   c2	1	-1	1	-1	1
   c3	2	0	-1	0	2
   c4	3	-1	0	1	-1
   c5	3	1	0	-1	-1
   r	4	2	1	0	0

7. Since splitting fields are determined up to isomorphism, we may as well assume that K $\subseteq$ $\mathbb {C}$. Since the roots of X⁴ - 2 are ±$\sqrt[4]{2}$, ±i$\sqrt[4]{2}$, we see that K = $\mathbb {Q}$([$\sqrt[4]{2}$], i). Now X⁴ - 2 is irreducible over $\mathbb {Q}$ by Eisenstein's criterion for the prime 2, so [$\mathbb {Q}$($\sqrt[4]{2}$) : $\mathbb {Q}$] = 4. Also i$\notin$$\mathbb {Q}$($\sqrt[4]{2}$) and i satisfies X² + 1 = 0, consequently [K : $\mathbb {Q}$($\sqrt[4]{2}$)] = 2. We deduce that [K : $\mathbb {Q}$] = 8 and therefore | Gal(K/$\mathbb {Q}$)| = 8. Now a group of order 8 has a normal subgroup of order 2 (a p-group has normal subgroups of any order dividing the order of the group); let H be a normal subgroup of order 2 in G, and let L be the fixed field of H. Then H has index 4 in G, so by the fundamental theorem of Galois theory, we see that L is a normal extension of degree 4 over $\mathbb {Q}$, as required.