Lecture 1.1: An introduction to groups

Transcription

1 Lectue.: An intoduction to goups Matthew Macauley Depatment o Mathematical Sciences Clemson Univesity Math 85, Abstact Algeba I M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I / 35

2 What is a goup? Deinition A nonempty set with an associative binay opeation is a semigoup. A semigoup S with an identity element such that x = x = x o all x S is a monoid. A goup is a monoid G with the popety that evey x G has an invese y G such that xy = yx =. Poposition. The identity o a monoid is unique. 2. Each element o a goup has a unique invese. 3. I x, y G, then (xy) = y x. Remaks I the binay opeation is addition, we wite the identity as. Easy to check that x m x n = x m+n and (x m ) n = x nm, m, n Z. [Additive analogue?] I xy = yx o all x, y G, then G is said to be abelian. In this lectue, we ll gain some intuition o goups beoe we begin a igoous mathematical teatment o them. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 2 / 35

3 Examples o goups. G = {, } R; multiplication. 2. G = Z, Q, R, C; addition. 3. G = Q = Q \ {}; multiplication. (Also woks o G = R, C, but not Z.) 4. G = Pem(S), the set o pemutations o S; unction composition. Special case: G = S n, the set o pemutations o S = {,..., n}. 5. D n = symmeties o a egula n-gon. 6. G = Q 8 = {±, ±i, ±j, ±k}, whee := I 4 4 and [ ] [ ] i =, j = Note that i 2 = j 2 = k 2 = ijk =., k = [ ]. 7. Klein 4-goup, i.e., the symmeties o a ectangle: {[ ] [ ] [ ] [ ]} V = {, v, h, } =,,, 8. Symmeties o a ieze diagam, wallpape, cystal, platonic solid, etc. Remak. Witing a goup G with matices is called a epesentation o G. (What ae some advantages o doing this?) M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 3 / 35

4 Cayley diagams A totally optional, but vey useul way to visualize goups, is using a Cayley diagam. This is a diected gaph (G, E), whee one ist ixes a geneating set S. We wite G = S. Then: Vetices: elements o G Diected edges: geneatos. The vetices can be labeled with elements, with coniguations, o unlabeled. Example. Two Cayley diagams o Z 6 = {,, 2, 3, 4, 5} = = 2, 3 : M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 4 / 35

5 The dihedal goup D The set D 3 =, o symmeties o an equilateal tiangle is a goup geneated by a clockwise 2 otation, and a hoizontal lip. 3 2 It can also be geneated by and anothe election g Hee ae two dieent Cayley diagams o D 3 =, =, g, whee g =. The ollowing ae seveal (o many!) pesentations o this goup: D 3 =, 3 = 2 =, = =, g 2 = g 2 = (g) 3 =. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 5 / 35

6 The quatenion goup The ollowing Cayley diagam, laid out two dieent ways, descibes a goup o size 8 called the quatenion goup, oten denoted Q 8 = {±, ±i, ±j, ±k}. i k j k j i j k i j i k The numbes j and k individually act like i =, because i 2 = j 2 = k 2 =. Multiplication o {±i, ±j, ±k} woks like the coss poduct o unit vectos in R 3 : ij = k, jk = i, ki = j, ji = k, kj = i, ik = j. Hee ae two possible pesentations o this goup: Q 8 = i, j, k i 2 = j 2 = k 2 = ijk = = i, j i 4 = j 4 =, iji = j. Recall that we can altenatvely espesent Q 8 with matices. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 6 / 35

7 The 7 types o ieze pattens Remaks The symmety goups o these ae geneated by some subset o the ollowing symmeties: t = tanslation, g = glide election, h = hoizontal election, v = vetical election, = 8 otation. These 7 symmetic goups all into 4 classes up to isomophism. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 7 / 35

8 The 7 types o wallpape pattens Fieze goups ae one-dimensional symmety goups. Two-dimensional symmety goups ae called wallpape goups. Thee ae 7 wallpapes goups, shown below, with the oicial IUC notation, adopted by the Intenational Union o Cystallogaphy in 952. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 8 / 35

9 Cystallogaphy Thee-dimensional symmety goups ae called cystal goups. Thee ae 23 cystal goups. One such cystal is shown below. The study o cystals is called cystallogaphy, and goup theoy plays a big ole is this banch o chemisty. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 9 / 35

10 Subgoups Deinition A subset H G that is a goup is called a subgoup o G, and denoted H G. Examples. What ae some o the subgoups o the goups we ve seen?. G = {, } R; multiplication. 2. G = Z, Q, R, C; addition. 3. G = Q = Q \ {}; multiplication. (Also woks o G = R, C, but not Z.) 4. G = Pem(S), the set o pemutations o S; unction composition. Special case: G = S n, the set o pemutations o S = {,..., n}. 5. D n = symmeties o a egula n-gon. 6. G = Q 8 = {±, ±i, ±j, ±k}, whee := I 4 4 and [ ] [ ] i =, j = Note that i 2 = j 2 = k 2 = ijk =., k = [ ]. 7. Klein 4-goup, i.e., the symmeties o a ectangle: {[ ] [ ] [ ] [ ]} V = {, v, h, } =,,, 8. Symmeties o a ieze diagam, wallpape, cystal, platonic solid, etc. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I / 35

11 Subgoups (poos done on the boad) Poposition.4 A nonempty set H G is a subgoup i and only i xy H o all x, y H. Coollay.5 I {H α} is any collection o subgoups o G, then H α G. α Evey set S G geneates a subgoup, denoted S. Thee ae two ways to think o this: om the bottom, up, as wods in S S, whee whee S = {x x S}: S = { x x 2 x k x i S S, k N } om the top, down: S := S H α G H α. Think o S as the smallest subgoup containing S. Poposition { x, x 2 x k x i S S, k N } = S H α G H α. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I / 35

12 Cyclic goups (poos done on the boad) Deinition A goup G is cyclic i G is geneated by a single element, i.e., i G = x. Examples (Z, +) = =. Rotational symmeties o a egula n-gon, C n :=. [O the additive goup (Z n, +).] Given x G, deine the ode o x to be x := x. Poposition.6 Suppose x = n < and x m =. Then n m. Poposition.7 Evey subgoup o a cyclic goup is cyclic. Coollay I G = x o ode n <, and k n, then x n/k is the unique subgoup o ode k in G. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 2 / 35

13 Cosets Deinition I H G and x, y G, then x and y ae conguent mod H, witten x y (mod H), i y x H. Conguent modulo H means the dieence o x and y lies in H. Easy execise: is an equivalence elation o any H. Remak x y (mod H) means x = yh o some h H. Deinition The equivalence class containing y is yh := {yh h H}, called the let coset o H containing y. Note that xh = yh (as sets) i x y (mod H). M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 3 / 35

14 Cosets Recall that o each x G, the let coset o H containing x is xh := {xh h H}. We can similaly deine the ight coset o H containing x as Hx := {hx h H}. H i ih i j Q 8 k D 3 D 3 jh j let cosets o H = also the ights cosets o H k kh the let coset the ight coset Notice that the let and ight cosets o the subgoup H = D 3 ae dieent: H H H H H H M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 4 / 35

15 Cosets The index o H in G, denoted [G : H] is the numbe o distinct let cosets o H in G. Lagange s theoem I H G, then G = [G : H] H. Deinition The nomalize o H in G, denoted N G (H), is N G (H) = {g G : gh = Hg} = {g G : ghg = H}. It is easy to check that H N G (G) G. In the catoon below, the nomalize consists o the elements in the ed cosets. H g 2H g 3H... g nh H Hg 2. Hg n Hg 3 Patition o G by the let cosets o H Patition o G by the ight cosets o H M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 5 / 35

16 Nomal subgoups Deinition A subgoup H G is nomal i gh = Hg o all g G. We wite H G. Useul emak (execise) The ollowing conditions ae all equivalent to a subgoup H G being nomal: (i) gh = Hg o all g G; ( let cosets ae ight cosets ); (ii) ghg = H o all g G; ( only one conjugate subgoup ) (iii) ghg H o all g G; ( closed unde conjugation ). (iv) N G (H) = G ( evey element nomalizes H ). Big idea (execise) I N G, then thee is a well-deined quotient goup: G/N := {xn x G}, xn yn := xyn. I G is witten additively, then cosets have the om x + N, and (x + N) + (y + N) = (x + y) + N. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 6 / 35

17 Nomal subgoups and quotients Deinition The cente o G is the set Z(G) := {x G xy = yx o all y G}. It is easy to show that Z(G) G. Example. The cente o Q 8 is N =. Let s see what the natual quotient η : Q 8 Q 8 /N looks like in tems o Cayley diagams. i i N i i in N in Q 8 Q 8 Q 8 /N j j k Q 8 oganized by the subgoup N = k jn j j k let cosets o N ae nea each othe Do you notice any elationship between Q 8 / Ke(φ) and Im(φ)? k kn jn kn collapse cosets into single nodes M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 7 / 35

18 A visual intepetation o the quotient map being well-deined Let s ty to gain moe insight. Conside a goup G with subgoup H. Recall that: each let coset gh is the set o nodes that the H-aows can each om g (which looks like a copy o H at g); each ight coset Hg is the set o nodes that the g-aows can each om H. The ollowing igue depicts the potential ambiguity that may aise when cosets ae collapsed. g 2H g H g 3H blue aows go om g H to multiple let cosets collapse cosets g 2H g H g 3H ambiguous blue aows g H g 2H collapse cosets blue aows go om g H to a unique let coset g H g 2H unambiguous blue aows The action o the blue aows above illustates multiplication o a let coset on the ight by some element. That is, the pictue shows how let and ight cosets inteact. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 8 / 35

19 Homomophisms Deinition A homomophism is a unction : G H such that (xy) = (x)g(y) o all x, y G. I is, it is a monomophism. I is onto, it is an epimomophism. I is and onto, it is an isomophism. We say that G and H ae isomophic, and wite G = H. A homomophism : G G is an endomophism. An isomophism : G G is an automophism. The kenel o a homomophism : G H is the set ke = {x G (x) = }. Poposition I : G H is a homomophism, then ke is a subgoup o G, and is i and only i ke = {}. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 9 / 35

20 Homomophisms Examples.. Let N G. Then η : G G/N, whee η : g gn is a homomophism called the natual quotient. 2. Let G = (R, +), H = { R > }. Then : G H, () = e is an isomophism. The invese map is : H G, (x) = ln x. (Veiy this!) 3. Let G = D 3, H = {, }. Deine (x) = Then is a homomophism. (Check!) 4. Let G be abelian and n Z. Then { x is a otation x is a election : G G, (x) = x n is an endomophism, since (xy) n = x n y n. 5. Let G = S 3, H = Z 6. Then G = H. (Why?) M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 2 / 35

21 Automophisms Poposition The set Aut(G) o automohpisms o G is a goup with espect to composition. Remaks. An automophism is detemined by whee it sends the geneatos. An automophism φ must send geneatos to geneatos. In paticula, i G is cyclic, then it detemines a pemutation o the set o (all possible) geneatos. Examples. Thee ae two automophisms o Z: the identity, and the mapping n n. Thus, Aut(Z) = C Thee is an automophism φ: Z 5 Z 5 o each choice o φ() {, 2, 3, 4}. Thus, Aut(Z 5 ) = C 4 o V 4. (Which one?) 3. An automophism φ o V 4 = h, v is detemined by the image o h and v. Thee ae 3 choices o φ(h), and then 2 choices o φ(v). Thus, Aut(V 4 ) = 6, so it is eithe C 6 = C2 C 3, o S 3. (Which one?) M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 2 / 35

22 Automophism goups o Z n Deinition The multiplicative goup o integes modulo n, denoted Z n o U(n), is the goup U(n) := {k Z n gcd(n, k) = } whee the binay opeation is multiplication, modulo n. U(5) = {, 2, 3, 4} = C4 U(8) = {, 3, 5, 7} = C2 C U(6) = {, 5} = C Poposition The automophism goup o Z n is Aut(Z n) = {σ a a U(n)} = U(n), whee σ a : Z n Z n, σ a() = a. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 22 / 35

23 Automophisms o D 3 Let s ind all automophisms o D 3 =,. We ll see a vey simila example to this when we study Galois theoy. Clealy, evey automophism φ is completely detemined by φ() and φ( ). Since automophisms peseve ode, i φ Aut(D 3 ), then φ(e) = e, φ() = o }{{ 2, } φ( ) =,, o. }{{} 2 choices 3 choices Thus, thee ae at most 2 3 = 6 automophisms o D 3. Let s ty to deine two maps, (i) α: D 3 D 3 ixing, and (ii) β : D 3 D 3 ixing : { α() = α( ) = { β() = β( ) = I claim that: these both deine automophisms (check this!) these geneate six dieent automophisms, and thus α, β = Aut(D 3 ). To detemine what goup this is isomophic to, ind these six automophisms, and make a goup pesentation and/o multiplication table. Is it abelian? M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 23 / 35

24 Automophisms o D 3 An automophism can be thought o as a e-wiing o the Cayley diagam. id β α αβ α 2 α2 β M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 24 / 35

25 Automophisms o D 3 Hee is the multiplication table and Cayley diagam o Aut(D 3 ) = α, β. id α α 2 β αβ α 2 β id id α α 2 β αβ α 2 β id α α 2 β αβ α 2 β α α 2 id α 2 β β αβ α 2 id α αβ α 2 β β β αβ α 2 β id α α 2 αβ α 2 β β α 2 id α α 2 β β αβ α α 2 id It is puely coincidence that Aut(D 3 ) = D 3. Fo example, we ve aleady seen that Aut(Z 5 ) = U(5) = Z 4, Aut(Z 6 ) = U(6) = Z 2, Aut(Z 8 ) = U(8) = Z 2 Z 2. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 25 / 35

26 Automophisms o V 4 = h, v The ollowing pemutations ae both automophisms: α : h v hv and β : h v hv h id h h h β v h v v v h hv hv v hv hv hv v hv h α v h h αβ h h v hv v hv hv h v hv hv v v hv h α2 hv h h α2 β hv h v h v v hv v v hv hv h v hv M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 26 / 35

27 Automophisms o V 4 = h, v Hee is the multiplication table and Cayley diagam o Aut(V 4 ) = α, β = S 3 = D3. id α α 2 β αβ α 2 β id id α α 2 β αβ α 2 β id α α 2 β αβ α 2 β α α 2 id α 2 β β αβ α 2 id α αβ α 2 β β β αβ α 2 β id α α 2 αβ α 2 β β α 2 id α α 2 β β αβ α α 2 id Note that α and β can be thought o as the pemutations h v hv and h v hv and so Aut(G) Pem(G) = S n always holds. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 27 / 35

28 The ist isomophism theoem Fundamental homomophism theoem (FHT) I φ: G H is a homomophism, then Im(φ) = G/ Ke(φ). The FHT says that evey homomophism can be decomposed into two steps: (i) quotient out by the kenel, and then (ii) elabel the nodes via φ. G (Ke φ G) φ any homomophism Im φ quotient pocess q i emaining isomophism ( elabeling ) G / Ke φ goup o cosets Poo Constuct an explicit map i : G/ Ke(φ) Im(φ) and pove that it is an isomophism... M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 28 / 35

29 The ist isomophism theoem Fundamental homomophism theoem (FHT) I φ: G H is a homomophism, then Im(φ) = G/ Ke(φ). Let s evist a amilia example to illustate this. Conside a homomophism: It is easy to check that Ke(φ) = Q 8. φ: Q 8 V 4, φ(i) = h, φ(j) = v. The FHT says that this homomophism can be done in two steps: (i) quotient by, and then (ii) elabel the nodes accodingly. N i i in N in h Q 8 Q 8 /N Q 8 /N jn j j k let cosets o N = k kn jn kn collapse cosets into single nodes v hv elabel nodes into single nodes M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 29 / 35

30 A pictue o the isomophism i : Z 2 Z/ 2 (om the VGT website) M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 3 / 35

31 How to show two goups ae isomophic The standad way to show G = H is to constuct an isomophism φ: G H. When the domain is a quotient, thee is anothe method, due to the FHT. Useul technique Suppose we want to show that G/N = H. Thee ae two appoaches: (i) Deine a map φ: G/N H and pove that it is well-deined, a homomophism, and a bijection. (ii) Deine a map φ: G H and pove that it is a homomophism, a sujection (onto), and that Ke φ = N. Usually, Method (ii) is easie. Showing well-deinedness and injectivity can be ticky. Fo example, each o the ollowing ae esults that we will see vey soon, o which (ii) woks quite well: Z/ n = Z n; Q / = Q + ; AB/B = A/(A B) G/(A B) = (G/A) (G/B) (assuming A, B G); (assuming G = AB). M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 3 / 35

32 The Second Isomophism Theoem Diamond isomophism theoem Let H G, and N N G (H). Then (i) The poduct HN = {hn h H, n N} is a subgoup o G. (ii) The intesection H N is a nomal subgoup o G. (iii) The ollowing quotient goups ae isomophic: HN/N = H/(H N) H G HN H N N Poo (sketch) Deine the ollowing map φ: H HN/N, φ: h hn. I we can show:. φ is a homomophism, 2. φ is sujective (onto), 3. Ke φ = H N, then the esult will ollow immediately om the FHT. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 32 / 35

33 The Thid Isomophism Theoem Feshman theoem Conside a chain N H G o nomal subgoups o G. Then. The quotient H/N is a nomal subgoup o G/N; 2. The ollowing quotients ae isomophic: (G/N)/(H/N) = G/H. G G/N H N H/N (G/N) (H/N) = G H (Thanks to Zach Teitle o Boise State o the concept and gaphic!) M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 33 / 35

34 The Fouth Isomophism Theoem Coespondence theoem Let N G. Thee is a coespondence between subgoups o G/N and subgoups o G that contain N. In paticula, evey subgoup o G/N has the om A := A/N o some A satisying N A G. This means that the coesponding subgoup lattices ae identical in stuctue. Example Q 8 Q 8 / V 4 i j k i / j / k / h vh v / The quotient Q 8 / is isomophic to V 4. The subgoup lattices can be visualized by collapsing to the identity. M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 34 / 35

35 Coespondence theoem (ull vesion) Let N G. Then thee is a bijection om the subgoups o G/N and subgoups o G that contain N. In paticula, evey subgoup o G/N has the om A := A/N o some A satisying N A G. Moeove, i A, B G, then. A B i and only i A B, 2. I A B, then [B : A] = [B : A], 3. A, B = A, B, 4. A B = A B, 5. A G i and only i A G. Example D 4 D 4 / V 4,,, / /, / h vh v 3 / M. Macauley (Clemson) Lectue.: An intoduction to goups Math 85, Abstact Algeba I 35 / 35