PRELIMINARY EXAMINATION IN ALGEBRA August 13, 2001

Let \(G\) be a finite group, and let \(H\) be a subgroup of index \(p\), where \(p\) is the smallest prime number dividing the order of \(G\). Show that \(H\) is a normal subgroup of \(G\).

Show that there are no simple groups of order 132.

Show that a polynomial \(f(X)\) of degree \(n\) over the field \(\mathbb {Z}/p\mathbb {Z}\) is irreducible if and only if \(f(X)\) is relatively prime to the polynomial \(X^{p^m}-X\) for all \(m\) less than or equal to \(n/2\).

Show that there exists a Galois extension of the rational numbers with Galois group \(\mathbb {Z}/7\mathbb {Z}\).

Let \(R\) be a unique factorization domain with quotient field \(K\). Let \(f(X)\) be a polynomial in \(R[X]\). Show that if \(f(X)\) can be factored into polynomials of lower degree in \(K[X]\), then \(f(X)\) can be factored into polynomials of lower degree in \(R[X]\).

Let \(G\) be a finite group. Prove the following version of Schur’s Lemma: if \(\rho \colon G\to M_n(\mathbb {C})\) is a finite dimensional irreducible representation of \(G\) over the complex numbers, then every element of \(M_n(\mathbb {C})\) that commutes with \(\rho (g)\) for all \(g\in G\) is a scalar matrix.

Let \(N\) be a positive integer. Prove that the natural map from \(\mathrm {SL}_2(\mathbb {Z})\) to \(\mathrm {SL}_2(\mathbb {Z}/N\mathbb {Z})\) is surjective.

Let \(G\) be a nonabelian group of order \(p^3\) for a prime \(p\) and and let \(Z\) be its center. Show that \(G/Z\cong \mathbb {Z}/p\mathbb {Z}\oplus \mathbb {Z}/p\mathbb {Z}\).

Show that \(\mathbb {Q}\otimes _{\mathbb {Z}}\mathbb {Q}\cong \mathbb {Q}\), and that \(\mathbb {Q}\otimes \mathbb {Z}/N\mathbb {Z}=0\) for every positive integer \(N\), while \(\mathbb {Q}\otimes \left (\prod \mathbb {Z}/N\mathbb {Z}\right )\ne 0\), where the product is taken over all positive integers \(N\).

Show that if a 3 by 3 matrix with entries in \(\mathbb {Q}\) satisfies \(A^8=I\), then it satisfies \(A^4=I\).

Let \(G\) be a finite abelian group and let \(n\) be an integer. Show that the map which sends \(g\) to \(ng\) is a group homomorphism and that it is an isomorphism if and only if \(n\) is relatively prime to the order of \(G\).