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Abstract Algebra I

by: Myrtice Robel

Abstract Algebra I MATH 5550

Myrtice Robel
GPA 3.63


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This 8 page Class Notes was uploaded by Myrtice Robel on Tuesday October 27, 2015. The Class Notes belongs to MATH 5550 at University of Wyoming taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/230315/math-5550-university-of-wyoming in Mathematics (M) at University of Wyoming.


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Date Created: 10/27/15
ReVIew Baslc Terminology and Results Also Group Actions Handout Janualy 12 2009 A muitzplzcamE omup ls a set G on whlch multlpllcatlon ls de ned and closed so that cc 6 G f0 all M e G such that l F0 all AW 6 G we have awe ma lll F0 all a e G thele elnsts aquot e G such that cat Theze exlsls 1 E G such that fox all GD 6 G we have 1 a G1 1 The opelatlon we have denoted by jmtapnmtwn l e slmply wlltlng the elements a and y next to each othel as ln aoy n ot el examples of gloups luxtaposltlon may be leplaced by a symbol such as an o etc fo the blnaly opelatlon A bmaw opmtcon on G ls a ma G X G a G l e takl eyely oldeled pall of elements of G to an element ofG thus closule ls lmpllcltly lequlled Thus a 970sz ls a set G wlth a blnaly opelatlon whlch ls assoclatlve f0 w lch has an ldentlty element and eyely element of G has an lnvelse ln gzoups ale taken to be multlpllcatlve unless othelwlse specl ed gxoup G ls obelcun If 393 ya f0 all a y e G The and lGl The 39 such that M 1 0 If no such lntegel k exlsts the oldel of a ls m mts and we wllte lal be A subwasz of G ls a subset g G whlch ls a gloup If the blnaly opelatlon of G ls lestllcted to H ln thls case we wllte H g G We know that a nonempty subset of G ls a subgzoup lll lt ls closed undel Laklrg ploducts and lnvelses Note that the lntelsectlon of two subgzoups of G ol any collectlon of subgzoups of G ls agaln a subgzoup of G The mmul subwasz 1 g G ls usually denoted slmply as 1 1 l en a subset S g G we denote by S 1ang1e srang1e ln TEX the unlque smallest subgzoup of G contalnlng s thls conslsts of 1 togethel wlth all elements of s m of G ls lts caldlnallty 1 and thellvallols u la 1 1 We call 3 the subgng gensmtsd by s In paltlculal If s G we say that G ls osnmtal by s A cycltc omup ls a gzoup genelated by a slrgle element thus a cycllc gzoup ls elthel l a llnlte cycllc gzoup Ofoldel a l e 1 m as an whele atquot 1ol u an lnllnlte cycllc gzoup ch 711cc2a 1 In both cases note that loci so the two meanings of order7 are compatible the order of an element is the order of the subgroup that it generates The centralizer of an element 9 E G is the subgroup Ggg ac E G avg git More generally if S Q G is any subset the centralizer of S in G is the subgroup GgSEszssx for allsE S A right coset of a subgroup H g G is a subset of the form Hac has h E H for some it E G All the right cosets of H have cardinality lHl and they partition the set G This proves Lagrange s Theorem lGllGiHllHl where G H denotes the index of H in G ie the number of right cosets of H in G In particular if G is finite then every subgroup of G has order dividing One could just as well use left cosets icH och h E H in place of right cosets giving another partition of G A subgroup K g G is normal denoted K 31 G if the left and right cosets of K coincide icK Kai for all it E G In this case Kacy for all gay 6 G and it is not hard to see that the set of cosets GK Kai it E G is a group called the quotient group obeyK lfH GandK GthenHK G hereHKhkh EH he lfHK 31 G then HK 31 G If icH Hie we say that it normalizes H The normalizer of H in G is the set of elements of G which normalize H ie NgH ac E G icH Note that H 31 NgH g G and we have the equality NgH G iff H 31 G The centre of G is the subgroup ZG z E G zg 92 for all g E G Clearly ZG 31 G and equality holds iff G is abelian If G and H are groups a homomorphism from G to H is a map 6 G a H such that Wavy 6x6y for all any 6 G In this case the image is a subgroup 6G g H and the kernel is a normal subgroup ker6l g E G 69 1 31 G In fact every normal subgroup is the kernel of some homomorphism so a normal subgroup is the same thing as the kernel of a homomorphism Indeed if K 31 G then K is the kernel of the canonical homomorphism G a GK ac gt gt Kai A group G is simple if its only normal subgroups are the trivial subgroup l and G itself A Herculean effort by dozens of mathematicians has led to the classification of finite simple groups CFSG The proof of this result the longest written proof in the history of mathematics occupies tens of thousands of pages The list of finite simple groups includes several infinite families and 26 sporadic groups the largest of which the Monster has order 808017424794512875886459904961710757005754368000000000 2 A homomorphism 6 G a H is one to one iff its kernel is the trivial subgroup l A homomorphism 6 is called an isomorphism if it is bijective If there exists an isomorphism 6 G a H then we say G is isomorphic to H and we write G g H This relation between groups of isomorphism is an equivalence relation lsomorphism preserves all the intrinsic properties of a group such as whether a group is abelian the number of elements of a given order etc We have the First Isomorphism Theorem lf 6 G a H is any group homomorphism then G ker6 g 6G The Second Isomorphism Theorem asserts that if H g G and K 31 G then K Note that H K 31 H also HK is a subgroup of G containing K so that K 31 The Third Isomorphism Theorem asserts that if K 31 G and L is another normal subgroup of G contained in K so that in particular L 31 K then GLKL g GK Note that KL 1 GL An automorphism of G is an isomorphism from G to itself The automorphisms of G form a group under composition denoted AutG Given 9 E G define wg G a G by 119 gicg l Then 119 is bijective its inverse is 1971 and 119 w gxg lgyg l game so 119 is an automorphism Note that 119 is the identity map on G iff g E ZG Automorphisms of the form wg are called inner automorphisms not of this form are called outer It is easily checked that 119 0 14 119 and 119 1 11971 thus the inner automorphisms of G form a subgroup lnnG Aut G In fact this is a normal subgroup lnnG 1 Aut G The commutator of two elements it y E G is the element y p ly lacy E G Note that y 1 iff it and y commute For two subgroups H and K we denote HKlthk heH mm The derived subgroup of G is the normal subgroup G G G 31 G The quotient GG is abelian and is called the obeliohizotioh of G In fact G is the unique smallest normal subgroup whose quotient is abelian The external direct product of two groups H and K is the group HgtltKhk hEH kEK with cornponentwise multiplication Here we may identify H and K with H gtlt l and l gtlt K respectively so we have a group H gtlt K having two normal subgroups with trivial intersection but whose product is the entire group H gtlt K Likewise if G is any group having norrnal subgroups H and K such that H K l and HK G then we say G is the internal direct product of H and K and we write G H gtlt K The distinction between internal and external direct products is merely a matter of viewpoint hence we have essentially one notion of a direct product of groups Every abelian group is a direct product of cyclic groups Let p be prirne A p group is a group of order pk for some k 2 1 An elementary abeliau p group is a direct product of cyclic groups of order p where p is prime A p subgroup of a finite group G is a subgroup H g G such that H is a p group A Sylou p subgroup of G is a p subgroup H g G whose index G H is not divisible by p A subgroup H g G is characteristic denoted H4 G if 6H H for every 6 E Aut G Every characteristic subgroup is normal but not conversely We say G is characteristically simple if its only characteristic subgroups are the trivial subgroup l and G itself Every such group is a direct product of isornorphic sirnple groups Group Actions If X is any set the set of permutations of X ie bijections X a X forms a group under composition This group denoted Sme is the symmetric group on X When X is a finite set we often take X l23 n in which case Sme is denoted S7 the symmetric group of degree rt A permutation group is a subgroup G g Sme for some X in this case the degree of the permutation group is the cardinality Given g E G and it E X we write g ac gt gt 9 Some authors write instead it gt gt Below we explain the differences between these two notational conventions The orbit of a point it E G is the subset Gx9g G X The orbits of G on X form a partition of X If there is only one orbit ie X itself we say G is trausitiue The stabilizer of a point it E X is the subgroup Gmg Gac9acltG The size of an orbit is the index of the corresponding stabilizer local G Goal 4 More generally let X be any set and let G be a group We say that G acts on X if there is a rule that assigns to every 9 E G a permutation of X denoted in gt gt 9 as above in such a way that x9h icgh for all it E X and g h E G The reason this is slightly more general than the situation described above is that we sometimes allow distinct elements of G to give the same permutation of X More precisely an action or representation of G on X is a homomorphism 6 G a Sme Here each 9 E G gives a permutation 69 X a X it gt gt 3699 We write simply 9 in place of 3599 if the choice of action 6 is clear from context lf 6 is one to one we say the action 6 is faithful in this case we may identify G with the permutation group 6G g Sme and we are back in the situation described above But in general we define orbits and stabilizers for any group action just as we do for permutation groups Ezrample Fractional Linear Transformations Let X C U 00 the one point compactification of the complex numbers We refer to X as the Riemann sphere or as the complea projective line Let G GL2CC the group of all invertible 2 gtlt 2 complex matrices Consider the action of G on X by fractional linear transformations the matrix 9 2 maps 6 iszCCandbzd7 0 bzd ZHZQ oo iszCCandbzd0 ifzoo The exceptional values of the fractional linear transformation X a X are chosen so as to ensure continuity with respect to the topology of X which is that of the Riemann sphere Here an open neighbourhood of 00 E X is simply the complementin X of a closed bounded subset of C This action is transitive ie every point of X can be mapped to every other point by some fractional linear transformation The stabilizer of 00 is the subgroup consisting of lower triangular matrices 2 with ad 31 0 the stabilizer of 0 is the subgroup consisting of upper triangular matrices g S with ad 31 0 The action is not faithful since if A is any nonzero complex number then the matrices a b d Aa Ab c d an Ac Ad give the same fractional linear transformation In fact the kernel of the action of G on X the set of all matrices fixing every point of X is just the set of scalar matrices AI 3 g for A 31 0 These matrices comprise the centre ZG and so GZG PGL2C is the group of permutations induced by G Example The Regular Action Let G be any group Then G acts on G itself by left multiplication the element g E G gives the map A9 G a G at gt gt gx Note that A9 is bijective its inverse is A971 and that Ag 0 Ah Agh so we do indeed have an action of G on G This action is transitive and the stabilizer of any point it E G is the trivial subgroup 1 so the action is faithful We may therefore regard G as a permutation group on G This gives Gayley s Representation Theorem every group is isomorphic to a permutation group And for a finite group G we see that G is isomorphic to a permutation group of degree n lGj hence G is isomorphic to a subgroup of S7 The action g gt gt A9 is called the left regular action ofG on G We may similarly define the right regular action of G on G if we are careful to distinguish between left and right actions see comments below Example Conjugation Let a group G act on G itself by conjugation Thus each g E G gives the permutation 119 G a G at gt gt gzg l The orbits of this action are simply the conjugacy classes of G The stabilizer of it E G is simply its centralizer Og From the formula for the orbit size we have W G 2 0am Thus every conjugacy class has size dividing the group order it is in fact the index of the corresponding centralizer The kernel of the action of G on itself by conjugation is ZG In particular the action is faithful iff G has trivial centre Left and Right Actions Let G be a group and X a set A left action of G on X is a map G gtlt X a X gz gt gt gac such that QWWDWMW for all g h E G and it E X We may write gx in place of gin A right action of G on X is a map X gtlt G a X zg gt gt cg such that whm for all g h E G at E X It is natural in this case to write 9 in place of mg The essential notational difference here is that for left actions function composition is performed right to left for right actions function composition is performed left to right Expressed in this way the distinction between left and right actions seems a rather minor notational issue 6 Left actions are typically preferred in undergraduate courses where the experience with function notation that most students acquire comes from pre calculus and calculus courses At the more advanced levels left actions are generally favored by analysts and right actions are generally favored by algebraists However one often wants to consider both left and right actions in the same context and then we cannot really learning about the subtle distinction Moreover because different authors have different personal preferences it is best for us to confront this issue head on As a first example let us denote by F1X71 the space of row vectors of length 11 over a field F and by F71 the space of column vectors of length n The group GLnF consisting of invertible n gtlt n matrices over F acts naturally on F1X71 on the right it also acts on F71X1 on the left For another example any group G naturally acts on itself by both left and right multi plication We have defined the left regular representation of G on itself as the permutation action where g E G acts on G by way of the permutation Ag G a G x gt gt gas The associative property for G then yields gh 90196 gh gh so that Agh Ag 0 Ah as expected We try to do the same for right multiplication given 9 E G a naive first attempt to define right multiplication by 9 would be pg G a G x gt gt avg We check that Paltph my 019 MM so that pg 0 pk pkg rather than pgh The failure to obtain pgh here can be expressed by saying that the map 9 gt gt pg is not a homomorphism or equivalently the usual right multiplication of G on G does not satisfy the usual rules for a left action where composition of maps is right to left as in undergraduate calculus courses We will remedy this situation by instead defining 71 pga 9 We now check that Pg 0 Ph pgh as before In fact this trick can be used to turn any right action into a left action and vice 71 versa Note that the conjugation action 119 gang simply becomes 1amp9 Ag 0 pg pg 0 Ag in this setting Although we have defined conjugation as a left action with right to left composition 119 0 14 wgh sometimes it is preferable to use instead right action 9 g lxg 7 so that 9h acgh And why would right action ever be preferable to left action We show how cycle notation for Sn works using both conventions and how left action has at least two problems which are resolved using right actions Let7s consider the composition of 243576 with 1523 using both conventions If we use left action then we must compose right to left thus 243576l523 176543 To find the image of each point of X 1 2 7 under this composite permutation requires careful Zig Zagging back and forth for example we check that 1 gt gt 7 under the composite map 24357f6l 23 Using the standard representation of functions acting on the left we may equivalently write 24357615231 2435765 7 Here the notation is ambiguous as the expressions 1 and 5 are easily misinterpreted as cycles of length 1 Both of these dif culties disappear with right action 11523243576 5243576 7 Here the left to right composition of cycles is consistent with the left to right interpretation of each cycle thus i5 23243gf6 Reference G Eric Moorhouse Abstract Algebra I course notes for Math 5550 httpwwwuwyo edumoorhousehandoutsalgebrapdf 932 KB


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