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Determine, up to isomorphism, the groups of order 30.

Let \(G\) be a finite simple group with identity \(e\). Suppose that \(A\) and \(B\) are distinct maximal proper subgroups of \(G\). If \(A\) and \(B\) are abelian, prove that \(A\cap B=\{e\}\).

Hint: Prove that \(A\cap B\) is normal in \(G\).

Suppose that \(p\) is the smallest prime dividing the order of a group \(G\), and that \(P\) is a Sylow \(p\)-subgroup of \(G\). If \(P\) is cyclic, show that \[ N_G(P)=C_G(P), \] i.e., that the normalizer of \(P\) in \(G\) agrees with the centralizer of \(P\) in \(G\).

Find all solutions of \(x^2=1\) in the ring \(\mathbb {Z}/91\).

Consider the ideal \(I=(2,1+\sqrt {-5})\) in the ring \(R=\mathbb {Z}[\sqrt {-5}]\). Is \(I\) a prime ideal? Is \(I\) a projective \(R\)-module?

Let \(A=\mathbb {F}_3[x]\), i.e., \(A\) is a polynomial ring in one variable over the field with 3 elements. Suppose \(M\) and \(N\) are finitely generated \(A\)-modules such that \[ M\oplus \frac {A}{x^3+1} \cong N\oplus \frac {A}{x+1}\oplus \frac {A}{x+1}\oplus \frac {A}{x+1}. \] Are \(M\) and \(N\) isomorphic?

Let \(R=\mathbb {Z}[i]\) be the ring of Gaussian integers. Consider the \(R\)-module \(M\) generated by two elements \(x\) and \(y\), subject to the relations \(ix+2y=0\) and \(2x-iy=0\). How many elements does \(M\) have?

Let \(R\) be the ring \(\mathbb {Q}[x,y]\), and let \(I\) be the ideal \(I=(x,y)\). What is the \(\mathbb {Q}\)-vector space rank of \(\operatorname {Tor}^{R}_{1}(R/I,I)\)?

Determine the extension degree of the splitting field of \(x^7-1\) over \(\mathbb {F}_{11}\).

Determine the Galois group of \(x^6+3\) over \(\mathbb {Q}\).