7 problems.
- A group with a subgroup of index \(n\) has a normal subgroup \(N\) with \(n\leq[G:N]\leq n!\), and there is no simple group of order \(36\)
- \(x^p-t\) is irreducible over \(\mathbb{F}_p(t)\) and splits as a \(p\)-th power in its splitting field
- Recall that a ring \(A\) is called if it admits a direct
- Rings with \(a^2=a\): Jacobson radical, characteristic, commutativity, and finite rings isomorphic to \((\ZZ/2\ZZ)^d\)
- \(\Gal(\overline{\mathbb{F}_p}/\mathbb{F}_p)\) has no nontrivial finite subgroups
- Rational irreps of \(C_p\) of dimensions \(1\) and \(p-1\), and abelian groups whose rational representations are absolutely irreducible
- Class sums form a \(\mathbb{Z}\)-basis of the center of \(\mathbb{Z}[G]\), and \(|C|\chi_\pi(C)/\dim V\) is an algebraic integer