Fall 2016 Problem 2: Let G be a finite group that acts transitively on a set X of cardinality ≥ 2. Show that there exists an element g ∈ G which acts on X without any fixed points. Is the same true if G is infinite?

Spring 2016 Problem 4: Prove that every finite group G of order > 2 has a nontrivial automorphism.

Fall 2017 Problem 6: Prove that if p is a prime number, then every group G with p^{2} elements is abelian.

Fall 2001 Problem 2:

(1) Give an example of a group and a subroup which is not normal.

(2) Show that every group of order 33 has a normal subgroup of order 11.

Winter 2002 Problem 9: Assume that H and K are normal subgroups of G with trivial intersection.

(1) Show that if h ∈ H and k ∈ K then hk = kh.

(2) Show that if h_{i} ∈ H and k_{i} ∈ K with h_{1}k_{1} = h_{2}k_{2} then h_{1} = h_{2} and k_{1} = k_{2}.

(3) Show that HK = {hk|h ∈ H, k ∈ K} is a group isomorphic to H × K.

Fall 2014 Problem 2: Show that every subgroup of index equal to the smallest prime dividing the order of a group in normal.

Fall 2002 Problem 2 Consider a finite group G with subgroup S and normalizer N = N_{G}(S).

(1) Show that N is a subgroup of G.

(2) Show that if S ≤ K ≤ G are groups then S is normal in K iff K ≤ N.

(3) Show that the number of subgroups of G conjugate to S is the index [G : N].

(4) Find N_{S_{4}}(⟨(1, 2, 3, 4)⟩).