1. In a group G, consider the subset K of all commutators aba⁻¹b⁻¹ where a,b are elements of G. Let C be the intersection of all subgroups containing K. (In other words, it is the smallest subgroup containing K.)
(a) Prove that C is normal, and that the quotient group G/C is abelian.
(b) Prove a homomorphism of G to an abelian group contains C in its kernel. Derive from this that every homomorphism to an abelian group A can be represented as composition G -> G/C -> A, where the first arrow is the standard projection to the quotient group.
(c) Find the commutator subgroup in S₄.
2. In a group G, let Z be the subset of all those elements which commute with all elements from G.
(a) Prove that Z is a normal subgroup (it is called the center of G), and that G is abelian if and only if Z=G.
(b) Let p be prime. Prove that every group of order p² is abelian. ( Hint: Express the number of conjugacy classes in G using Cauchy's counting formula applied to the action of G on itself by conjugations.)
3. Show that the 8 unit quaternions {1,-1, i,-i, j,-j, k, -k} form a group with respect to multiplication (recall that i^2=i^2=k^2=-1, ij=k-ji, jk=i=-kj, ki=j=-ik), and compute the center and the commutator subgroups in this group.

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Abelian Groups Problems
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