DOCSLIB.ORG

2. Normal Subgroup and Quotient Group We Begin by Stating a Couple of Elementary Lemmas

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
2. Normal Subgroup and Quotient Group We Begin by Stating a Couple of Elementary Lemmas 2. normal subgroup and quotient group We begin by stating a couple of elementary lemmas. 2.1. Lemma. Let A and B be sets and f : A → B be an onto function. For b ∈ B,recall that f −1(b)={a ∈ A: f(a)=b}.LetF = {f −1(b): b ∈ B}.ThenF is a partition of A and there is a natural one to one correspondence between B and F given by b #→ f −1(b). 2.2. Lemma. Let A and B be subsets of a group G.Ifg,h ∈ G,theng(hA)=(gh)A.If A ⊆ B and g ∈ G,thengA ⊆ gB. The proofs of the two lemmas above will be left as exercises. We often use easy results like these without mentioning explicitly. 2.3. Lemma. Let φ : G → H be a group homomorphism and let N =ker(φ).Then φ−1(φ(g)) = gN = Ng for all g ∈ G. Proof. Take g ∈ G.Letx ∈ φ−1(φ(g)) or in other words, φ(x)=φ(g). Then −1 −1 −1 φ(g x)=φ(g) φ(x)=φ(g) φ(g)=eH, so g−1x ∈ N, or in other words x ∈ gN.Thusφ−1(φ(g)) ⊆ gN. Conversely, if x ∈ gN, −1 then x = gn for some n ∈ N,soφ(x)=φ(g)φ(n)=φ(g)eH = φ(g), so x ∈ φ (φ(g)). Thus gN ⊆ φ−1(φ(g)). This proves φ−1(φ(g)) = gN. A similar argument shows that φ−1(φ(g)) = Ng (Exercise: Write out this argument). ! 2.4. Definition. Let G be a group. A subgroup N of G is called a normal subgroup if gN = Ng for all g ∈ G. In this case we need not distinguish between left and right cosets. So we simply talk of cosets of N. 2.5. Lemma. Let G be a group and N be a subgroup of G.Thefollowingareequivalent. (a) gN = Ng for all g ∈ G. (b) gNg−1 = N for all g ∈ G. (c) If g ∈ G and n ∈ N,thengng−1 ∈ N. Proof. Suppose (a) holds. Then gNg−1 =(gN)g−1 =(Ng)g−1 = N(gg−1)=N.Thus(a) implies (b). Clearly (b) implies (c). Assume (c). Let x ∈ gN.Thenx ∈ gn for some n ∈ N. So x = gng−1g ∈ Ng since gng−1 ∈ N by our assumption. Thus gN ⊆ Ng. Similarly show that Ng ⊆ gN.SogN = Ng. Thus (c) implies (a). ! 2.6 (Motivating the construction of a quotient). Lemma 2.3 says that kernels of ho- momorphisms are normal subgroups. Conversely, we shall see that all normal subgroups appear as kernels of homomorphisms. Let G be a group and let N be a normal subgroup of G. We shall construct a onto group homomorphism π : G → Q such that ker(π)=N. This group Q will be called the quotient of G by the normal subgroup N and will be denoted by Q = G/N. To motivate the construction, suppose we have an onto group homomorphism π : G → Q and let ker(π)=N.Letq ∈ Q. Since π is onto, we can pick an element g ∈ G such that π(g)=q. Then Then lemma 2.3 gives us π−1(q)=gN. In words, this says that the preimages of elements of Q are just the cosets of N. So the elements of Q are in one to one correspondence with the set of cosets of N in G. Also verify that if q1 and q2 are two elements −1 −1 −1 of Q such that π (q1)=g1N and π (q2)=g2N,thenπ (q1q2)=g1g2N (Exercise: verify this). This suggests the definition that follows. 3 2.7. Definition. Let G be a group and N be a normal subgroup. Let G/N denote the set of cosets of N in G.LetC and D be two cosets of N in G. We say that an element c ∈ G is a representative of the coset C if c ∈ C. Choose a representatitve c of the coset C and a representative d of the coset D. Claim: The coset cdN only depends on the cosets C and D and does not depend on the choice of the representatives c and d. Proof. Let c1 be another representative of C and d1 be another representative of D.Then −1 −1 −1 −1 c1c ∈ N and d1d ∈ N. Since N is normal, c(d1d )c ∈ N as well. So −1 −1 −1 −1 −1 −1 (c1d1)(cd) = c1d1d c =(c1c )c(d1d )c ∈ N. So c1d1 ∈ Ncd = cdN.Soc1d1N = cdN. ! Define a binary operation on G/N as follows: Given two cosets C and D,wechoosecoset representatives c and d of C and D respectively and let let (C.D)tobethecosetcdN.The claim we just proved imply that this procedure gives a well defined binary operation on G/N. Let us reiterate the definition of the binary operation in slightly different words. If c ∈ C and d ∈ D,thenC = cN and D = dN. So our definition says (cN).(dN)=cdN. Let us also reiterate that we have already argued this is a well defined operation on G/N and this argument required the fact that N is a normal subgroup. Next one verifies that G/N with the binary operation just defined becomes a group where the identity is the coset N and inverse of gN is g−1N. This group G/N is called the quotient of G by N. There is a canonical onto function from π : G → G/N given by π(g)=gN. Verify that π is a homomorphism and ker(π)=N. 2.8. Example. Verify that Zn = Z/nZ. 3. Constructing homomorphisms 3.1. Lemma. Let G be a group and a ∈ G.Thenthereexistsahomomorphismφ : Z → G given by φ(n)=an for all n ∈ Z.Thisφ is the unique homomorphism from Z to G such that φ(1) = a. sketch of proof. Define φ(n)=an for all n ∈ Z. Verify that φ : Z → G is a homomorphism such that φ(1) = a.Letφ′ : Z → G be any homomorphisms such that φ′(1) = a.Then φ′(n)=an = φ(n) for all n ∈ Z.Soφ′ = φ. ! The above lemma says that constructing homomorphisms from Z to any group is easy. You can send 1 to anything you want and the homomorphism is completely determined once you decide where 1 goes. 3.2. Corollary. The map φ(k)=k mod n is the unique homomorphism φ : Z → Zn such that φ(1) = 1 mod n.ThekernelofthishomomorphismisnZ. 3.3. Lemma. Let φ1 : G → H1 and φ2 : G → H2 be two homomorphisms. Then there is a homomorphism φ : G → H1 × H2 given by φ(G)=(φ1(G),φ2(G)). 4 3.4. Theorem. Let A, B, C be three groups and let f : A → B and g : A → C be two homomorphisms. (i) if there exists a homomorphism h : B → C such that h ◦ f = g,thenker(f) ⊂ ker(g). (ii) Suppose f is onto and ker(f) ⊂ ker(g). (a) Then there exists a unique homomorphism h : B → C such that h ◦ f = g. (b) If g is onto, then so is h. (c) If ker(g)=ker(f),thenh is one to one. (d) If g is onto and ker(g)=ker(f),thenh is an isomorphism. Proof. (i) Exercise. (ii) (a) Uniqueness: Suppose h, h′ be two homomorphisms such that h ◦ f = g = h′ ◦ f. Pick b ∈ B. Since f is onto there exists a ∈ A such that f(a)=b.thenh(b)=h(f(a)) = g(a)=h′(f(a)) = h′(b). Fix b ∈ B.Supposea, a′ ∈ A such that f(a)=f(a′)=b,soh = h′. Thus there can be atmost one h with the given properties. Existence: Let b ∈ B.Supposea, a′ ∈ A such that f(a)=f(a′)=b.Thenf(a−a′)=0, so (a − a′) ∈ ker(f), so (a − a′) ∈ ker(g), so g(a − a′)=0,sog(a)=g(a′). Thus we observe that “g takes the same value on all the elements of A that map to b”. Given b ∈ B,wecan pick a ∈ A such that f(a)=b (since f is onto) and then define h(b)=g(a). Because of our observation, this procedure gives us a well defined function h : B → C. Verify that h is a homomorphism. Now suppose a ∈ A.Letb = f(a). So a is an element of A such that f(a)=b.Soh(b)=g(a), or h(f(a)) = g(a). So h ◦ f = g. This proves part (a). (b) Now suppose g is onto. Let c ∈ C. Then there exists a ∈ A such that g(a)=c.It follows that h(f(a)) = c,soh is onto. This proves part (b). (c) Now suppose g is onto and ker(f)=ker(g). To show h is one to one, it is enough to verify that ker(h)=eB.Letb ∈ ker(h), that is, h(b)=eC . Pick a ∈ A such that f(a)=b. Then g(a)=h(f(a)) = h(b)=eC ,soa ∈ ker(g)=ker(f), so b = f(a)=eB.Soker(h)=eB. This proves part (c). Part (d) follows from part (b) and (c). ! 3.5. Corollary. Let m, n be positive integers. Then there exists a unique homomorphism h : Zmn → Zm such that h(1 mod mn)=1modm. sketch of proof. Let f : Z → Zmn and g : Z → Zm be the homomorphisms f(k)=k mod mn and g(k)=k mod m.Thenker(f)=mnZ ⊆ mZ =ker(g).
Load more

Recommended publications
  • Quotient Groups §7.8 (Ed.2), 8.4 (Ed.2): Quotient Groups and Homomorphisms, Part I
    Math 371 Lecture #33 x7.7 (Ed.2), 8.3 (Ed.2): Quotient Groups x7.8 (Ed.2), 8.4 (Ed.2): Quotient Groups and Homomorphisms, Part I Recall that to make the set of right cosets of a subgroup K in a group G into a group under the binary operation (or product) (Ka)(Kb) = K(ab) requires that K be a normal subgroup of G. For a normal subgroup N of a group G, we denote the set of right cosets of N in G by G=N (read G mod N). Theorem 8.12. For a normal subgroup N of a group G, the product (Na)(Nb) = N(ab) on G=N is well-defined. We saw the proof of this before. Theorem 8.13. For a normal subgroup N of a group G, we have (1) G=N is a group under the product (Na)(Nb) = N(ab), (2) when jGj < 1 that jG=Nj = jGj=jNj, and (3) when G is abelian, so is G=N. Proof. (1) We know that the product is well-defined. The identity element of G=N is Ne because for all a 2 G we have (Ne)(Na) = N(ae) = Na; (Na)(Ne) = N(ae) = Na: The inverse of Na is Na−1 because (Na)(Na−1) = N(aa−1) = Ne; (Na−1)(Na) = N(a−1a) = Ne: Associativity of the product in G=N follows from the associativity in G: [(Na)(Nb)](Nc) = (N(ab))(Nc) = N((ab)c) = N(a(bc)) = (N(a))(N(bc)) = (N(a))[(N(b))(N(c))]: This establishes G=N as a group.
    [Show full text]
  • Group Homomorphisms
    1-17-2018 Group Homomorphisms Here are the operation tables for two groups of order 4: · 1 a a2 + 0 1 2 1 1 a a2 0 0 1 2 a a a2 1 1 1 2 0 a2 a2 1 a 2 2 0 1 There is an obvious sense in which these two groups are “the same”: You can get the second table from the first by replacing 0 with 1, 1 with a, and 2 with a2. When are two groups the same? You might think of saying that two groups are the same if you can get one group’s table from the other by substitution, as above. However, there are problems with this. In the first place, it might be very difficult to check — imagine having to write down a multiplication table for a group of order 256! In the second place, it’s not clear what a “multiplication table” is if a group is infinite. One way to implement a substitution is to use a function. In a sense, a function is a thing which “substitutes” its output for its input. I’ll define what it means for two groups to be “the same” by using certain kinds of functions between groups. These functions are called group homomorphisms; a special kind of homomorphism, called an isomorphism, will be used to define “sameness” for groups. Definition. Let G and H be groups. A homomorphism from G to H is a function f : G → H such that f(x · y)= f(x) · f(y) forall x,y ∈ G.
    [Show full text]
  • Categories of Sets with a Group Action
    Categories of sets with a group action Bachelor Thesis of Joris Weimar under supervision of Professor S.J. Edixhoven Mathematisch Instituut, Universiteit Leiden Leiden, 13 June 2008 Contents 1 Introduction 1 1.1 Abstract . .1 1.2 Working method . .1 1.2.1 Notation . .1 2 Categories 3 2.1 Basics . .3 2.1.1 Functors . .4 2.1.2 Natural transformations . .5 2.2 Categorical constructions . .6 2.2.1 Products and coproducts . .6 2.2.2 Fibered products and fibered coproducts . .9 3 An equivalence of categories 13 3.1 G-sets . 13 3.2 Covering spaces . 15 3.2.1 The fundamental group . 15 3.2.2 Covering spaces and the homotopy lifting property . 16 3.2.3 Induced homomorphisms . 18 3.2.4 Classifying covering spaces through the fundamental group . 19 3.3 The equivalence . 24 3.3.1 The functors . 25 4 Applications and examples 31 4.1 Automorphisms and recovering the fundamental group . 31 4.2 The Seifert-van Kampen theorem . 32 4.2.1 The categories C1, C2, and πP -Set ................... 33 4.2.2 The functors . 34 4.2.3 Example . 36 Bibliography 38 Index 40 iii 1 Introduction 1.1 Abstract In the 40s, Mac Lane and Eilenberg introduced categories. Although by some referred to as abstract nonsense, the idea of categories allows one to talk about mathematical objects and their relationions in a general setting. Its origins lie in the field of algebraic topology, one of the topics that will be explored in this thesis. First, a concise introduction to categories will be given.
    [Show full text]
  • Lie Group Structures on Quotient Groups and Universal Complexifications for Infinite-Dimensional Lie Groups
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Functional Analysis 194, 347–409 (2002) doi:10.1006/jfan.2002.3942 Lie Group Structures on Quotient Groups and Universal Complexifications for Infinite-Dimensional Lie Groups Helge Glo¨ ckner1 Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918 E-mail: [email protected] Communicated by L. Gross Received September 4, 2001; revised November 15, 2001; accepted December 16, 2001 We characterize the existence of Lie group structures on quotient groups and the existence of universal complexifications for the class of Baker–Campbell–Hausdorff (BCH–) Lie groups, which subsumes all Banach–Lie groups and ‘‘linear’’ direct limit r r Lie groups, as well as the mapping groups CK ðM; GÞ :¼fg 2 C ðM; GÞ : gjM=K ¼ 1g; for every BCH–Lie group G; second countable finite-dimensional smooth manifold M; compact subset SK of M; and 04r41: Also the corresponding test function r r # groups D ðM; GÞ¼ K CK ðM; GÞ are BCH–Lie groups. 2002 Elsevier Science (USA) 0. INTRODUCTION It is a well-known fact in the theory of Banach–Lie groups that the topological quotient group G=N of a real Banach–Lie group G by a closed normal Lie subgroup N is a Banach–Lie group provided LðNÞ is complemented in LðGÞ as a topological vector space [7, 30]. Only recently, it was observed that the hypothesis that LðNÞ be complemented can be omitted [17, Theorem II.2]. This strengthened result is then used in [17] to characterize those real Banach–Lie groups which possess a universal complexification in the category of complex Banach–Lie groups.
    [Show full text]
  • The Category of Generalized Lie Groups 283
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 199, 1974 THE CATEGORYOF GENERALIZEDLIE GROUPS BY SU-SHING CHEN AND RICHARD W. YOH ABSTRACT. We consider the category r of generalized Lie groups. A generalized Lie group is a topological group G such that the set LG = Hom(R, G) of continuous homomorphisms from the reals R into G has certain Lie algebra and locally convex topological vector space structures. The full subcategory rr of f-bounded (r positive real number) generalized Lie groups is shown to be left complete. The class of locally compact groups is contained in T. Various prop- erties of generalized Lie groups G and their locally convex topological Lie algebras LG = Horn (R, G) are investigated. 1. Introduction. The category FT of finite dimensional Lie groups does not admit arbitrary Cartesian products. Therefore we shall enlarge the category £r to the category T of generalized Lie groups introduced by K. Hofmann in [ll]^1) The category T contains well-known subcategories, such as the category of compact groups [11], the category of locally compact groups (see §5) and the category of finite dimensional De groups. The Lie algebra LG = Horn (R, G) of a generalized Lie group G is a vital and important device just as in the case of finite dimensional Lie groups. Thus we consider the category SI of locally convex topological Lie algebras (§2). The full subcategory Í2r of /--bounded locally convex topological Lie algebras of £2 is roughly speaking the family of all objects on whose open semiballs of radius r the Campbell-Hausdorff formula holds.
    [Show full text]
  • Självständiga Arbeten I Matematik
    SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Geometric interpretation of non-associative composition algebras av Fredrik Cumlin 2020 - No K13 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM Geometric interpretation of non-associative composition algebras Fredrik Cumlin Självständigt arbete i matematik 15 högskolepoäng, grundnivå Handledare: Wushi Goldring 2020 Abstract This paper aims to discuss the connection between non-associative composition algebra and geometry. It will first recall the notion of an algebra, and investigate the properties of an algebra together with a com- position norm. The composition norm will induce a law on the algebra, which is stated as the composition law. This law is then used to derive the multiplication and conjugation laws, where the last is also known as convolution. These laws are then used to prove Hurwitz’s celebrated the- orem concerning the different finite composition algebras. More properties of composition algebras will be covered, in order to look at the structure of the quaternions H and octonions O. The famous Fano plane will be the finishing touch of the relationship between the standard orthogonal vectors which construct the octonions. Lastly, the notion of invertible maps in relation to invertible loops will be covered, to later show the connection between 8 dimensional rotations − and multiplication of unit octonions. 2 Contents 1 Algebra 4 1.1 The multiplication laws . .6 1.2 The conjugation laws . .7 1.3 Dickson double . .8 1.4 Hurwitz’s theorem . 11 2 Properties of composition algebras 14 2.1 The left-, right- and bi-multiplication maps . 16 2.2 Basic properties of quaternions and octonions .
    [Show full text]
  • Quotient-Rings.Pdf
    4-21-2018 Quotient Rings Let R be a ring, and let I be a (two-sided) ideal. Considering just the operation of addition, R is a group and I is a subgroup. In fact, since R is an abelian group under addition, I is a normal subgroup, and R the quotient group is defined. Addition of cosets is defined by adding coset representatives: I (a + I)+(b + I)=(a + b)+ I. The zero coset is 0+ I = I, and the additive inverse of a coset is given by −(a + I)=(−a)+ I. R However, R also comes with a multiplication, and it’s natural to ask whether you can turn into a I ring by multiplying coset representatives: (a + I) · (b + I)= ab + I. I need to check that that this operation is well-defined, and that the ring axioms are satisfied. In fact, everything works, and you’ll see in the proof that it depends on the fact that I is an ideal. Specifically, it depends on the fact that I is closed under multiplication by elements of R. R By the way, I’ll sometimes write “ ” and sometimes “R/I”; they mean the same thing. I Theorem. If I is a two-sided ideal in a ring R, then R/I has the structure of a ring under coset addition and multiplication. Proof. Suppose that I is a two-sided ideal in R. Let r, s ∈ I. Coset addition is well-defined, because R is an abelian group and I a normal subgroup under addition. I proved that coset addition was well-defined when I constructed quotient groups.
    [Show full text]
  • LECTURE 12: LIE GROUPS and THEIR LIE ALGEBRAS 1. Lie
    LECTURE 12: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups Definition 1.1. A Lie group G is a smooth manifold equipped with a group structure so that the group multiplication µ : G × G ! G; (g1; g2) 7! g1 · g2 is a smooth map. Example. Here are some basic examples: • Rn, considered as a group under addition. • R∗ = R − f0g, considered as a group under multiplication. • S1, Considered as a group under multiplication. • Linear Lie groups GL(n; R), SL(n; R), O(n) etc. • If M and N are Lie groups, so is their product M × N. Remarks. (1) (Hilbert's 5th problem, [Gleason and Montgomery-Zippin, 1950's]) Any topological group whose underlying space is a topological manifold is a Lie group. (2) Not every smooth manifold admits a Lie group structure. For example, the only spheres that admit a Lie group structure are S0, S1 and S3; among all the compact 2 dimensional surfaces the only one that admits a Lie group structure is T 2 = S1 × S1. (3) Here are two simple topological constraints for a manifold to be a Lie group: • If G is a Lie group, then TG is a trivial bundle. n { Proof: We identify TeG = R . The vector bundle isomorphism is given by φ : G × TeG ! T G; φ(x; ξ) = (x; dLx(ξ)) • If G is a Lie group, then π1(G) is an abelian group. { Proof: Suppose α1, α2 2 π1(G). Define α : [0; 1] × [0; 1] ! G by α(t1; t2) = α1(t1) · α2(t2). Then along the bottom edge followed by the right edge we have the composition α1 ◦ α2, where ◦ is the product of loops in the fundamental group, while along the left edge followed by the top edge we get α2 ◦ α1.
    [Show full text]
  • Math 412. Adventure Sheet on Quotient Groups
    Math 412. Adventure sheet on Quotient Groups Fix an arbitrary group (G; ◦). DEFINITION: A subgroup N of G is normal if for all g 2 G, the left and right N-cosets gN and Ng are the same subsets of G. NOTATION: If H ⊆ G is any subgroup, then G=H denotes the set of left cosets of H in G. Its elements are sets denoted gH where g 2 G. The cardinality of G=H is called the index of H in G. DEFINITION/THEOREM 8.13: Let N be a normal subgroup of G. Then there is a well-defined binary operation on the set G=N defined as follows: G=N × G=N ! G=N g1N ? g2N = (g1 ◦ g2)N making G=N into a group. We call this the quotient group “G modulo N”. A. WARMUP: Define the sign map: Sn ! {±1g σ 7! 1 if σ is even; σ 7! −1 if σ is odd. (1) Prove that sign map is a group homomorphism. (2) Use the sign map to give a different proof that An is a normal subgroup of Sn for all n. (3) Describe the An-cosets of Sn. Make a table to describe the quotient group structure Sn=An. What is the identity element? B. OPERATIONS ON COSETS: Let (G; ◦) be a group and let N ⊆ G be a normal subgroup. (1) Take arbitrary ng 2 Ng. Prove that there exists n0 2 N such that ng = gn0. (2) Take any x 2 g1N and any y 2 g2N. Prove that xy 2 g1g2N.
    [Show full text]
  • Right-Angled Artin Groups in the C Diffeomorphism Group of the Real Line
    Right-angled Artin groups in the C∞ diffeomorphism group of the real line HYUNGRYUL BAIK, SANG-HYUN KIM, AND THOMAS KOBERDA Abstract. We prove that every right-angled Artin group embeds into the C∞ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG ∞ group, embeds into the C diffeomorphism group of the real line. 1. Introduction The right-angled Artin group on a finite simplicial graph Γ is the following group presentation: A(Γ) = hV (Γ) | [vi, vj] = 1 if and only if {vi, vj}∈ E(Γ)i. Here, V (Γ) and E(Γ) denote the vertex set and the edge set of Γ, respec- tively. ∞ For a smooth oriented manifold X, we let Diff+ (X) denote the group of orientation preserving C∞ diffeomorphisms on X. A group G is said to embed into another group H if there is an injective group homormophism G → H. Our main result is the following. ∞ R Theorem 1. Every right-angled Artin group embeds into Diff+ ( ). Recall that a finitely generated group G is a limit group (or a fully resid- ually free group) if for each finite set F ⊂ G, there exists a homomorphism φF from G to a nonabelian free group such that φF is an injection when restricted to F . The class of limit groups fits into a larger class of groups, which we call residually RAAG groups. A group G is in this class if for each arXiv:1404.5559v4 [math.GT] 16 Feb 2015 1 6= g ∈ G, there exists a graph Γg and a homomorphism φg : G → A(Γg) such that φg(g) 6= 1.
    [Show full text]
  • MA4E0 Lie Groups
    MA4E0 Lie Groups December 5, 2012 Contents 0 Topological Groups 1 1 Manifolds 2 2 Lie Groups 3 3 The Lie Algebra 6 4 The Tangent space 8 5 The Lie Algebra of a Lie Group 10 6 Integrating Vector fields and the exponential map 13 7 Lie Subgroups 17 8 Continuous homomorphisms are smooth 22 9 Distributions and Frobenius Theorem 23 10 Lie Group topics 24 These notes are based on the 2012 MA4E0 Lie Groups course, taught by John Rawnsley, typeset by Matthew Egginton. No guarantee is given that they are accurate or applicable, but hopefully they will assist your study. Please report any errors, factual or typographical, to [email protected] i MA4E0 Lie Groups Lecture Notes Autumn 2012 0 Topological Groups Let G be a group and then write the group structure in terms of maps; multiplication becomes m : G × G ! G −1 defined by m(g1; g2) = g1g2 and inversion becomes i : G ! G defined by i(g) = g . If we suppose that there is a topology on G as a set given by a subset T ⊂ P (G) with the usual rules. We give G × G the product topology. Then we require that m and i are continuous maps. Then G with this topology is a topological group Examples include 1. G any group with the discrete topology 2. Rn with the Euclidean topology and m(x; y) = x + y and i(x) = −x 1 2 2 2 3. S = f(x; y) 2 R : x + y = 1g = fz 2 C : jzj = 1g with m(z1; z2) = z1z2 and i(z) =z ¯.
    [Show full text]
  • What Does a Lie Algebra Know About a Lie Group?
    WHAT DOES A LIE ALGEBRA KNOW ABOUT A LIE GROUP? HOLLY MANDEL Abstract. We define Lie groups and Lie algebras and show how invariant vector fields on a Lie group form a Lie algebra. We prove that this corre- spondence respects natural maps and discuss conditions under which it is a bijection. Finally, we introduce the exponential map and use it to write the Lie group operation as a function on its Lie algebra. Contents 1. Introduction 1 2. Lie Groups and Lie Algebras 2 3. Invariant Vector Fields 3 4. Induced Homomorphisms 5 5. A General Exponential Map 8 6. The Campbell-Baker-Hausdorff Formula 9 Acknowledgments 10 References 10 1. Introduction Lie groups provide a mathematical description of many naturally-occuring sym- metries. Though they take a variety of shapes, Lie groups are closely linked to linear objects called Lie algebras. In fact, there is a direct correspondence between these two concepts: simply-connected Lie groups are isomorphic exactly when their Lie algebras are isomorphic, and every finite-dimensional real or complex Lie algebra occurs as the Lie algebra of a simply-connected Lie group. To put it another way, a simply-connected Lie group is completely characterized by the small collection n2(n−1) of scalars that determine its Lie bracket, no more than 2 numbers for an n-dimensional Lie group. In this paper, we introduce the basic Lie group-Lie algebra correspondence. We first define the concepts of a Lie group and a Lie algebra and demonstrate how a certain set of functions on a Lie group has a natural Lie algebra structure.
    [Show full text]