Fall 2014 Problem 1: Let G_{1}, G_{2} be finite index subgroups of a group G. Show that the intersection G_{1} ∩ G_{2} also has finite index in G.

2008 Problem 1: Show that every finitely generated group contains only finitely many subgrouups of index n.

Fall 2004 Problem 2: Show that no infinite simple group has proper subgroups of finite index.

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Fall 2018 Problem 1: Show that a free group of rank ≥ 2 is not nilpotent.

Fall 2016 Problem 1: Let F_{n} be the free group on n generators with n ≥ 2. Prove that the center Z(F) of F is trivial.

Fall 2015 Problem 5: Show that two free groups are isomorphic if and only if they have equal ranks.

Fall 2004 Problem 3 and two other exams: Show that there are no nontrivial homomorphisms from ℚ to a finitely generated Abelian group.

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Winter 2004 Problem 11: Show that ℚ is not a nontrivial direct sum.

Winter 2005 Problem 3: Show that ℚ contains no proper subgroups of finite index.