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# Class Note for MATH 201 at GW

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This 4 page Class Notes was uploaded by an elite notetaker on Saturday February 7, 2015. The Class Notes belongs to a course at George Washington University taught by a professor in Fall. Since its upload, it has received 28 views.

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Date Created: 02/07/15
Math 201 Algebra I Notes De nition 1 Algebra The study of structure that comes from endowing operations on the set De nition 2 Group A group is a pair G7gp where G is a set and Lp is a binary operation Lp G X G a G which is associative and there is an element 1 E G such that Va 6 G7gp1x Lp1 a and W E 911 E GKWWZI soltyc1 Typically7 we write a b or ab for gpa7 b and w 1 for the such that Lpy Wyn 1 Example 1 Z gt a gt R gt c gt groups under addition Q0 gtlt1Ro gtltc 07 are groups under multiplication De nition 3 Abelian Group A group is abelian if its operation is commutative Example 2 Let S be a set PerrnS bijectionsS a S PermS is a group under function cornposition Composition gives a well de ned closed operation PermS gtlt PermS a PermS lf 8 lt 007 then lPermSl le is a set With 1X1 n then write Sn for PermX Sn is the symmetric group Example 3 33 write 123 for the underlying set The following is an example of cornposing two elements in 33 am w gt i Mi 2 Z 2 Z De nition 4 Gagleg Graph Have one vertex for each group elernent arrows indicating what action each group element has on each other one N N as as N N as N Example 4 Z7 G ro Z gto Z gto Z gto Z gto 2 Z 0 1 2 Thus 1 connects the entire group Example 5 Cayley Group on 33 1 123 132 23 r 22 Edge labeled are under the operation of 2339cb7blile all the other edges are under the operation of Example 6 CLAIR group of invertible n X n matrices with entries in IR under multiplica tion GL means general linear group Example 7 Let T be a regular n7gon in R2 centered at 00 Dgn the group of rigid symmetries ofT which at T as a set De nition 5 Dihedral Group The groups described in Example 7 are known as the Dihedral Groups Note Dgn can be Viewed as a subgroup of CLAIR The edge 17 2 can go to 17 272 7 37357 441 7 575 7 6 or you can ip it and it can goto 2717372747375747675 It turns our that 33 and D6 are the same group September 11 2006 De nition 6 Let H Q G then H is a subgroup ofG ifH is a group in its own right under the operation ofG This is denoted H S G Theorem 1 In a group G ifH Q G7H 9 then H S G ltgt VLy E Hy 1w E 2 Proof Assume H S G and xy 6 H7 then y 1 E H by the existence of inverses in and group and y lx E H by Closure in the group H Now assume the Vxy E H we have y lx E H Taking y to be x7 we get 1 x lx E H Taking x to be 1 and y to be arbitrary7 we see that y 1 1 E H Finally7 for any xy 6 H7 we know y 1 E H so that yx y 1 1x E H D Theorem 2 If Ha is a collection of subgroups of G then UHD S G This justi es this de nition De nition 7 IfG is a group and S Q G then the subgroup generated by S is H 3 GS Q We say that H 3 GS Q H De nition 8 A group G is cyclic means G for some x E G Usually we write Example 8 Z lt1gt 71gt Example 9 Model D2 in terms of functions on R2 Let p R2 a 1R2 denote rotation by Let 0 R2 a 1R2 denote re ections across y 0 Then D2 20 Note ltpgt 3 D2 and lt0gt S DZn D2n MOW lt72 WW 1gt Theorem 3 IfG is cyclic and H S G then H is also cyclic De nition 9 Relation Let S be a set a relation ofS is a subset R C S X S if xy E R then we say x is related to y Denoted x y De nition 10 Re exive R is re exive if Vx E Sxx E R De nition 11 Symmetric R is symmetric if Vxy E Sxy E R gt yx E R De nition 12 Transitive R is transitive if Vxy2 E Sxy E R y2 E R gt E R De nition 13 Eguivalence Relation If a relation is re exive symmetric and transitive then it is an equivalence relation De nition 14 Partition The collection S0 of subsets ofS is a partition ofS means S U and V04 SD 0 S5 91 Theorem 4 1 Every equivalence relation de nes a partition S0 ofS in which xy 6 SD ltgt 711 6 R 2 Every partition gives rise to an equivalence relation by the same de nition 3 lf partition Sq arises from an equivalence relation7 then we Will call each SDt an equivalence class Example 10 In Z pick m E Z and de ne aEbltgtmlaEb Note that mZ 3 Z so that a E b ltgt Eb a E mZ Group G and subgroup H S G de ne an equivalence relation by a E y ltgt y la E H Actually y la E H ltgt a E yH So a E y ltgt a E yH IfH S G then a coset ofH is a subset ofG of the form yH for some y E G Theorem 5 The cosets ofH in G partition G

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