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Unformatted text preview: MATH 74 QUIZ 3 REVIEW The following are the major topics: (1) Groups and Homomorphisms : Definitions, uniqueness of inverses, elementary properties of ho- momorphisms (e.g. ( x- 1 ) = ( x )- 1 ) with proofs, kernels of homomorphisms, abelian groups, order of a group. (2) Examples of Groups : Z /n Z and general cyclic groups, D n , S n , GL ( n ), SL ( n ), O ( n ), SO ( n ) (3) Subgroups and Cosets : Lemma for determining if a subset is a subgroup, multiplication modulo a subgroup H , left and right cosets, normal subgroups. (4) Quotient Groups : Quotient groups, coset multiplication, equivalence between surjective homo- morphisms and normal subgroups, first isomorphism theorem. (5) Equivalence Relations : Definition, equivalence classes, Theorem: a partition of a set X into mutually disjoint subsets is the same as an equivalence relation (statement only). (6) Lagranges Theorem : Both the statement and proof of the theorem, theorem applied to the order of group elements, (7) Conjugacy Classes and Cauchys Theorem : The equivalence relation of begin conjugate, con- jugacy classes, normalizers, the center of a group, the class equation, Cauchys theorem....
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