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Z4 is abelian. 9. If G is a group and { g1 , . . , gn } ⊂ G is a set of elements, define g1 , . . , g n ⊂ G α αi to be the set containing 1 and all possible products of strings gi 1 · · · gi k . where 1 k the αi are integers. Since the products of any two elements of g1 , . . , gn is also in g1 , . . , gn , this is a subgroup of G, called the subgroup generated by { g1 , . . , gn }. If G = g1 , . . , gn , we say that the group G is generated by g1 , . . , gn . A group that is generated by a single element is called cyclic.

If G is a cyclic group of order n and k is a positive integer with gcd(n, k) = 1, show that the function f:G → G g → gk is an isomorphism from G to itself. 4. 4. Subgroups and cosets We being by considering the relationship between a group and its subgroups. 1. If G is a group with a subgroup, H, we can define an equivalence relation on elements of G. x≡y (mod H ) if there exists an element h ∈ H such that x = y · h. The equivalence classes in G of elements under this relation are called left-cosets of H.

It is surjective if for every t ∈ T, there exists an s ∈ S such that f (s) = t — so the image of f is all of T. It is bijective if it is injective and surjective. The reader may wonder what all of this has to do with groups. 3. e. for all g1 , g2 ∈ G f ( g1 g2 ) = f ( g1 ) f ( g2 ) ∈ H The set of all elements g ∈ G such that f ( g) = 1 ∈ H is called the kernel of f , denoted ker f . If f is bijective, it is called an isomorphism and the groups G and H are said to be isomorphic. An isomorphism from a group to itself is called an automorphism.

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Abstract Algebra [draft] by Justin R. Smith

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