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Unformatted text preview: Exam II Review Sheet Math 4023 Section 1 The second exam will be on Wednesday, October 20, 2010. The syllabus will be Sections II.1, II.2, II.3, III.1, III.2, III.3, III.4 and III.8, plus the handout on cyclic groups. Of course, the material is cumulative, and the listed sections depend on earlier sections, which it is assumed that you still know. Following are some of the concepts and results you should know: Know what a relation on a set X is and the various properties of R contained in deflnition 1.4 on Page 29: re exive, symmetric, antisymmetric, and transitive. Know how to represent relations on flnite sets by means of the adjacency matrix M R (Page 34). Know the two special types of relations: partial order (that is R is re exive, antisymmetric, and transitive), and equivalence relation (that is, R is re exive, symmetric, and transitive). Know how to represent a partial order on a flnite set by means of the Hasse diagram (Page 37). Know the fundamental fact about an equivalence relation. Namely, an equivalence relation on set X determines a partition of X into disjoint sets called equivalence classes . (Theorem 3.7, Page 42). Know what the equivalence classes [ a ] R of an equivalence relation are? What is a semigroup? What is a group? What is the difierence between a semigroup and a group? Know examples of groups such as S n , D 4 , Z , Z n , cyclic groups. In particular, know what elements of these groups are and what the group operation is in each case. Know the order of each of the groups which is flnite. Know the cancelation rules in a group. For example, if ab = ac then b = c . Exponential rules in groups. For example, a m a n = a m + n , ( a m ) n = a mn , and ( ab ) 1 = b 1 a 1 . What does it mean to be a subgroup? Know the criterion to be a subgroup (Proposition 3.8) and how to use it to check that H is a subgroup of a group G . What is an abelian group? What is a cyclic group? What is a generator of a cyclic group? What is the condition for an element a 2 Z n to have a multiplicative inverse? (Answer: a and n should be relatively prime integers. When a has a multiplicative inverse, know how to flnd it using the Euclidean algorithm. What is the order of a group (denoted j G j )? What is the order o ( g ) of an element g 2 G ? ( o ( g ) is the smallest positive integer m such that g m = e .) If o ( g ) = m , then g n = e if and only if m j n . Every subgroup of a cyclic group is cyclic (Proposition 3.19). The left cosets of a subgroup H are the sets aH = f ah : h 2 H g . They are precisely the equivalence classes of the equivalence relation a H b () a 1 b 2 H . The set of left cosets of H in G is denoted G=H ....
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 Fall '09

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