Last time: isomorphism theorems. Let G be a group. 1. If ϕ : G Ñ H is a homomorphism of groups, then kerpϕq Ĳ G and

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Leonard Daniels
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1 Last time: isomorphism theorems Let G be a group. 1. If ϕ : G Ñ H is a homomorphism of groups, then kerpϕq Ĳ G and G{kerpϕq ϕpgq.

2 Last time: isomorphism theorems Let G be a group. 1. If ϕ : G Ñ H is a homomorphism of groups, then kerpϕq Ĳ G and G{kerpϕq ϕpgq. 2. Let A, B ď G and assume A ď N G pbq. Then AB{B A{pA X Bq (with appropriate statements about normality).

3 Last time: isomorphism theorems Let G be a group. 1. If ϕ : G Ñ H is a homomorphism of groups, then kerpϕq Ĳ G and G{kerpϕq ϕpgq. 2. Let A, B ď G and assume A ď N G pbq. Then AB{B A{pA X Bq (with appropriate statements about normality). 3. Let A, B Ĳ G with A ď B. Then B{A Ĳ G{A and pg{aq{pb{aq pg{bq.

4 Last time: isomorphism theorems Let G be a group. 1. If ϕ : G Ñ H is a homomorphism of groups, then kerpϕq Ĳ G and G{kerpϕq ϕpgq. 2. Let A, B ď G and assume A ď N G pbq. Then AB{B A{pA X Bq (with appropriate statements about normality). 3. Let A, B Ĳ G with A ď B. Then B{A Ĳ G{A and pg{aq{pb{aq pg{bq. 4. Today: Every subgroup of G{N comes from projecting a subgroup of G, and the containment, generation, normality, and index information pass through via π the way you want them to.

5 A comment on our proofs In proving the 3rd iso thm, given by if H, K Ĳ G, then pg{hq{pk{hq G{K, we studied the map ϕ : G{H Ñ G{K defined by gh ÞÑ gk.

6 A comment on our proofs In proving the 3rd iso thm, given by if H, K Ĳ G, then pg{hq{pk{hq G{K, we studied the map ϕ : G{H Ñ G{K defined by gh ÞÑ gk. The first step was to show this was well-defined (independent of choice of representative in gh).

7 A comment on our proofs In proving the 3rd iso thm, given by if H, K Ĳ G, then pg{hq{pk{hq G{K, we studied the map ϕ : G{H Ñ G{K defined by gh ÞÑ gk. The first step was to show this was well-defined (independent of choice of representative in gh). Another way: ϕ is basically the map Φ : G Ñ G{K defined by g ÞÑ gk, except split into two parts by π ϕ G G{H G{K Φ

8 General principle: homomorphisms on quotient groups Any map ϕ : G{N Ñ H is a map on cosets gn

9 General principle: homomorphisms on quotient groups Any map ϕ : G{N Ñ H is a map on cosets gn, so the image is determined by g itself.

10 General principle: homomorphisms on quotient groups Any map ϕ : G{N Ñ H is a map on cosets gn, so the image is determined by g itself. Essentially, ϕ pulls back to a map Φ : G Ñ H where Φpgq ϕpgnq.

11 General principle: homomorphisms on quotient groups Any map ϕ : G{N Ñ H is a map on cosets gn, so the image is determined by g itself. Essentially, ϕ pulls back to a map Φ : G Ñ H where Φpgq ϕpgnq. Then ϕ is well-defined

12 General principle: homomorphisms on quotient groups Any map ϕ : G{N Ñ H is a map on cosets gn, so the image is determined by g itself. Essentially, ϕ pulls back to a map Φ : G Ñ H where Φpgq ϕpgnq. Then if and only if ϕ is well-defined ϕpgnq ϕpg 1 Nq for all g 1 P gn

13 General principle: homomorphisms on quotient groups Any map ϕ : G{N Ñ H is a map on cosets gn, so the image is determined by g itself. Essentially, ϕ pulls back to a map Φ : G Ñ H where Φpgq ϕpgnq. Then if and only if if and only if ϕ is well-defined ϕpgnq ϕpg 1 Nq for all g 1 P gn Φpgq Φpg 1 q for all g 1 N P gn

14 General principle: homomorphisms on quotient groups Any map ϕ : G{N Ñ H is a map on cosets gn, so the image is determined by g itself. Essentially, ϕ pulls back to a map Φ : G Ñ H where Φpgq ϕpgnq. Then if and only if if and only if if and only if ϕ is well-defined ϕpgnq ϕpg 1 Nq for all g 1 P gn Φpgq Φpg 1 q for all g 1 N P gn N ď kerpφq.

15 General principle: homomorphisms on quotient groups Any map ϕ : G{N Ñ H is a map on cosets gn, so the image is determined by g itself. Essentially, ϕ pulls back to a map Φ : G Ñ H where Φpgq ϕpgnq. Then if and only if if and only if if and only if ϕ is well-defined ϕpgnq ϕpg 1 Nq for all g 1 P gn Φpgq Φpg 1 q for all g 1 N P gn N ď kerpφq. So to define a homomorphism ϕ : G{N Ñ H, you can instead define a homomorphism Φ : G Ñ H and check that N ď kerpφq.

16 General principle: homomorphisms on quotient groups Any map ϕ : G{N Ñ H is a map on cosets gn, so the image is determined by g itself. Essentially, ϕ pulls back to a map Φ : G Ñ H where Φpgq ϕpgnq. Then if and only if if and only if if and only if ϕ is well-defined ϕpgnq ϕpg 1 Nq for all g 1 P gn Φpgq Φpg 1 q for all g 1 N P gn N ď kerpφq. So to define a homomorphism ϕ : G{N Ñ H, you can instead define a homomorphism Φ : G Ñ H and check that N ď kerpφq. In this case, we say Φ factors through N and ϕ is the induced homomorphism on G{N.

17 General principle: homomorphisms on quotient groups Any map ϕ : G{N Ñ H is a map on cosets gn, so the image is determined by g itself. Essentially, ϕ pulls back to a map Φ : G Ñ H where Φpgq ϕpgnq. Then if and only if if and only if if and only if ϕ is well-defined ϕpgnq ϕpg 1 Nq for all g 1 P gn Φpgq Φpg 1 q for all g 1 N P gn N ď kerpφq. So to define a homomorphism ϕ : G{N Ñ H, you can instead define a homomorphism Φ : G Ñ H and check that N ď kerpφq. In this case, we say Φ factors through N and ϕ is the induced homomorphism on G{N. Pictorially, we say the following diagram commutes, meaning that following either path from G to H gives the same image: G π G{N Φ ϕ H

18 Fourth (lattice) isomorphism theorem Theorem Let N Ĳ G. There natural projection π : G Ñ G{N gives a bijection where Ā πpaq A{N. ta N ď A ď Gu ÐÑ tā Ā ď G{Nu

19 Fourth (lattice) isomorphism theorem Theorem Let N Ĳ G. There natural projection π : G Ñ G{N gives a bijection where Ā πpaq A{N. ta N ď A ď Gu ÐÑ tā Ā ď G{Nu For all N ď A, B ď G, this bijection additionally satisfies 1. A Ĳ G if and only if Ā Ĳ Ḡ, 2. A ď B if and only if Ā ď B, 3. if A ď B, then B : A B : Ā, 4. xa, By xā, By.

20 Lattice of D 16 t1, r, r 2, r 3, r 4, r 5, r 6, r 7, s, sr, sr 2, sr 3, sr 4, sr 5, sr 6, sr 7 u t1, r 2, r 4, r 6, s, sr 2, sr 4, sr 6 u t1, r, r 2, r 3, r 4, r 5, r 6, r 7 u t1, r 2, r 4, r 6, sr, sr 3, sr 5, sr 7 u t1, r 4, sr 2, sr 6 u t1, r 4, s, sr 4 u t1, r 2, r 4, r 6 u t1, r 4, sr 3, sr 7 u t1, r 4, sr, sr 5 u t1, sr 6 u t1, sr 2 u t1, sr 4 u t1, su t1, r 4 u t1, sr 3 u t1, sr 7 u t1, sr 5 u t1, sru t1u

21 Lattice of D 16 t1, r, r 2, r 3, r 4, r 5, r 6, r 7, s, sr, sr 2, sr 3, sr 4, sr 5, sr 6, sr 7 u t1, r 2, r 4, r 6, s, sr 2, sr 4, sr 6 u t1, r, r 2, r 3, r 4, r 5, r 6, r 7 u t1, r 2, r 4, r 6, sr, sr 3, sr 5, sr 7 u t1, r 4, sr 2, sr 6 u t1, r 4, s, sr 4 u t1, r 2, r 4, r 6 u t1, r 4, sr 3, sr 7 u t1, r 4, sr, sr 5 u t1, sr 6 u t1, sr 2 u t1, sr 4 u t1, su t1, r 4 u t1, sr 3 u t1, sr 7 u t1, sr 5 u t1, sru t1u

22 Lattice of D 16 t1, r, r 2, r 3, r 4, r 5, r 6, r 7, s, sr, sr 2, sr 3, sr 4, sr 5, sr 6, sr 7 u t1, r 2, r 4, r 6, s, sr 2, sr 4, sr 6 u t1, r, r 2, r 3, r 4, r 5, r 6, r 7 u t1, r 2, r 4, r 6, sr, sr 3, sr 5, sr 7 u t1, r 4, sr 2, sr 6 u t1, r 4, s, sr 4 u t1, r 2, r 4, r 6 u t1, r 4, sr 3, sr 7 u t1, r 4, sr, sr 5 u t1, sr 6 u t1, sr 2 u t1, sr 4 u t1, su t1, r 4 u t1, sr 3 u t1, sr 7 u t1, sr 5 u t1, sru t1u

23 Lattice of D 16 tz, rz, r 2 Z, r 3 Z, sz, srz, sr 2 Z, sr 3 Zu tz, r 2 Z, sz, sr 2 Zu tz, rz, r 2 Z, r 3 Zu tz, r 2 Z, srz, sr 3 Zu tz, sr 2 Zu tz, szu tz, r 2 Zu tz, sr 3 Zu tz, srzu Z

24 The Hölder program, a story Goal: Classify, up to isomorphism, all finite groups.

25 The Hölder program, a story Goal: Classify, up to isomorphism, all finite groups. Lattice isomorphism theorem: G{N is the group those structure describes the structure of G above N

26 The Hölder program, a story Goal: Classify, up to isomorphism, all finite groups. Lattice isomorphism theorem: G{N is the group those structure describes the structure of G above N This gives us an idea of how to use induction (on size) to simplify the study of finite groups.

27 The Hölder program, a story Goal: Classify, up to isomorphism, all finite groups. Lattice isomorphism theorem: G{N is the group those structure describes the structure of G above N This gives us an idea of how to use induction (on size) to simplify the study of finite groups. If N Ĳ G, then the structure of G is equivalent to (1) the structure of G{N, (2) the structure of N, and (3) how G is built from N and G{N

28 The Hölder program, a story Goal: Classify, up to isomorphism, all finite groups. Lattice isomorphism theorem: G{N is the group those structure describes the structure of G above N This gives us an idea of how to use induction (on size) to simplify the study of finite groups. If N Ĳ G, then the structure of G is equivalent to (1) the structure of G{N, (2) the structure of N, and (3) how G is built from N and G{N Base case: We say a group G is simple if G ą 1 and the only normal subgroups of G are G and 1.

29 The Hölder program, a story Goal: Classify, up to isomorphism, all finite groups. Lattice isomorphism theorem: G{N is the group those structure describes the structure of G above N This gives us an idea of how to use induction (on size) to simplify the study of finite groups. If N Ĳ G, then the structure of G is equivalent to (1) the structure of G{N, (2) the structure of N, and (3) how G is built from N and G{N Base case: We say a group G is simple if G ą 1 and the only normal subgroups of G are G and 1. Hölder program: 1. Classify all finite simple groups. 2. Find all ways of putting simple groups together.

30 The Hölder program, a story Goal: Classify, up to isomorphism, all finite groups. Lattice isomorphism theorem: G{N is the group those structure describes the structure of G above N This gives us an idea of how to use induction (on size) to simplify the study of finite groups. If N Ĳ G, then the structure of G is equivalent to (1) the structure of G{N, (2) the structure of N, and (3) how G is built from N and G{N Base case: We say a group G is simple if G ą 1 and the only normal subgroups of G are G and 1. Hölder program: 1. Classify all finite simple groups. 2. Find all ways of putting simple groups together.

31 Classifying finite simple groups Theorem Every finite simple group is isomorphic to one of 1. a group in one of 18 (infinite) families, or 2. one of 26 sporadic groups.

32 Classifying finite simple groups Theorem Every finite simple group is isomorphic to one of 1. a group in one of 18 (infinite) families, or 2. one of 26 sporadic groups. For example, A tz p p primeu is one of the 18 infinite families of finite simple groups.

33 Classifying finite simple groups Theorem Every finite simple group is isomorphic to one of 1. a group in one of 18 (infinite) families, or 2. one of 26 sporadic groups. For example, A tz p p primeu is one of the 18 infinite families of finite simple groups. Namely, if p is prime, then Z p is simple (by Lagrange); and if p p 1, then Z p fl Z p 1.

34 Classifying finite simple groups Theorem Every finite simple group is isomorphic to one of 1. a group in one of 18 (infinite) families, or 2. one of 26 sporadic groups. For example, A tz p p primeu is one of the 18 infinite families of finite simple groups. Namely, if p is prime, then Z p is simple (by Lagrange); and if p p 1, then Z p fl Z p 1. The sort of theorem that went into the classification is as follows: Theorem (Feit-Thompson). G Z p for some prime p. If G is simple and of odd order, then (The proof of this theorem took about 255 pages of sophisticated and technical mathematics. We won t do it.)

35 Classifying finite simple groups Theorem Every finite simple group is isomorphic to one of 1. a group in one of 18 (infinite) families, or 2. one of 26 sporadic groups. Another example of one of the 18 families is B tsl n pfq{zpsl n pfqq n P Z ě2, F is a finite fieldu.

36 Classifying finite simple groups Theorem Every finite simple group is isomorphic to one of 1. a group in one of 18 (infinite) families, or 2. one of 26 sporadic groups. Another example of one of the 18 families is B tsl n pfq{zpsl n pfqq n P Z ě2, F is a finite fieldu. Namely, SL n pfq tm P GL n pfq detpmq 1u is finite if F is finite.

37 Classifying finite simple groups Theorem Every finite simple group is isomorphic to one of 1. a group in one of 18 (infinite) families, or 2. one of 26 sporadic groups. Another example of one of the 18 families is B tsl n pfq{zpsl n pfqq n P Z ě2, F is a finite fieldu. Namely, SL n pfq tm P GL n pfq detpmq 1u is finite if F is finite. But SL n pfq is not simple, since ZpGq is always normal$ in G, and, λ & 1 λ 2 /. ZpSL n pfqq... λ i P F, λ 1 λ 2 λ n 1 % ˇ /- 1 if F ą 1. λ n

38 Classifying finite simple groups Theorem Every finite simple group is isomorphic to one of 1. a group in one of 18 (infinite) families, or 2. one of 26 sporadic groups. Another example of one of the 18 families is B tsl n pfq{zpsl n pfqq n P Z ě2, F is a finite fieldu. Namely, SL n pfq tm P GL n pfq detpmq 1u is finite if F is finite. But SL n pfq is not simple, since ZpGq is always normal$ in G, and, λ & 1 λ 2 /. ZpSL n pfqq... λ i P F, λ 1 λ 2 λ n 1 % ˇ /- 1 λ n if F ą 1. It turns out, though, that SL n pfq{zpsl n pfqq is simple.

39 Classifying finite simple groups Theorem Every finite simple group is isomorphic to one of 1. a group in one of 18 (infinite) families, or 2. one of 26 sporadic groups. Another example of one of the 18 families is B tsl n pfq{zpsl n pfqq n P Z ě2, F is a finite fieldu. Namely, SL n pfq tm P GL n pfq detpmq 1u is finite if F is finite. But SL n pfq is not simple, since ZpGq is always normal$ in G, and, λ & 1 λ 2 /. ZpSL n pfqq... λ i P F, λ 1 λ 2 λ n 1 % ˇ /- 1 λ n if F ą 1. It turns out, though, that SL n pfq{zpsl n pfqq is simple. And the groups in B are distinct for distinct n and F.

40 Putting simple groups together Let A and B groups. Putting A and B together means finding a group G with normal subgroups N such that G{N A and N B.

41 Putting simple groups together Let A and B groups. Putting A and B together means finding a group G with normal subgroups N such that G{N A and N B. Example: B 1 ˆ B Ĳ A ˆ B and pa ˆ Bq{p1 ˆ Bq A.

42 Putting simple groups together Let A and B groups. Putting A and B together means finding a group G with normal subgroups N such that G{N A and N B. Example: B 1 ˆ B Ĳ A ˆ B and pa ˆ Bq{p1 ˆ Bq A. Example: The lattices for Q 8 and D 8 are Q 8 xiy xjy xky xs, r 2 y D 8 xry xsr, r 2 y x 1y 1 xsy xsr 2 y xr 2 y xsry xsr 3 y 1

43 Putting simple groups together Let A and B groups. Putting A and B together means finding a group G with normal subgroups N such that G{N A and N B. Example: B 1 ˆ B Ĳ A ˆ B and pa ˆ Bq{p1 ˆ Bq A. Example: The lattices for Q 8 and D 8 are Q 8 xiy xjy xky xs, r 2 y D 8 xry xsr, r 2 y x 1y xsy xsr 2 y xr 2 y xsry xsr 3 y 1 1 Highlighting the sublattices corresponding to Q 8 {x 1y and D 8 {xr 2 y shows that they have the same lattice.

44 Putting simple groups together Let A and B groups. Putting A and B together means finding a group G with normal subgroups N such that G{N A and N B. Example: B 1 ˆ B Ĳ A ˆ B and pa ˆ Bq{p1 ˆ Bq A. Example: The lattices for Q 8 and D 8 are Q 8 xiy xjy xky xs, r 2 y D 8 xry xsr, r 2 y x 1y xsy xsr 2 y xr 2 y xsry xsr 3 y 1 1 Highlighting the sublattices corresponding to Q 8 {x 1y and D 8 {xr 2 y shows that they have the same lattice. In fact, Q 8 {x 1y Z 2 ˆ Z 2 D 8 {xr 2 y.

45 Putting simple groups together Let A and B groups. Putting A and B together means finding a group G with normal subgroups N such that G{N A and N B. Example: B 1 ˆ B Ĳ A ˆ B and pa ˆ Bq{p1 ˆ Bq A. Example: The lattices for Q 8 and D 8 are Q 8 xiy xjy xky xs, r 2 y D 8 xry xsr, r 2 y x 1y xsy xsr 2 y xr 2 y xsry xsr 3 y 1 1 Highlighting the sublattices corresponding to Q 8 {x 1y and D 8 {xr 2 y shows that they have the same lattice. In fact, Q 8 {x 1y Z 2 ˆ Z 2 D 8 {xr 2 y. Further, x 1y xr 2 y.

46 Putting simple groups together Let A and B groups. Putting A and B together means finding a group G with normal subgroups N such that G{N A and N B. Example: B 1 ˆ B Ĳ A ˆ B and pa ˆ Bq{p1 ˆ Bq A. Example: The lattices for Q 8 and D 8 are Q 8 xiy xjy xky xs, r 2 y D 8 xry xsr, r 2 y x 1y xsy xsr 2 y xr 2 y xsry xsr 3 y 1 1 Highlighting the sublattices corresponding to Q 8 {x 1y and D 8 {xr 2 y shows that they have the same lattice. In fact, Q 8 {x 1y Z 2 ˆ Z 2 D 8 {xr 2 y. Further, x 1y xr 2 y. So G{N G 1 {N 1 and N N 1 does not imply G G 1!

47 Putting simple groups together Let A and B groups. Putting A and B together means finding a group G with normal subgroups N such that G{N A and N B. Example: B 1 ˆ B Ĳ A ˆ B and pa ˆ Bq{p1 ˆ Bq A. Example: The lattices for Q 8 and D 8 are Q 8 xiy xjy xky xs, r 2 y D 8 xry xsr, r 2 y x 1y xsy xsr 2 y xr 2 y xsry xsr 3 y 1 1 Highlighting the sublattices corresponding to Q 8 {x 1y and D 8 {xr 2 y shows that they have the same lattice. In fact, Q 8 {x 1y Z 2 ˆ Z 2 D 8 {xr 2 y. Further, x 1y xr 2 y. So G{N G 1 {N 1 and N N 1 does not imply G G 1! Point: There s more than one way to put A and B together.

Last time: Recall that the fibers of a map ϕ : X Ñ Y are the sets in ϕ 1 pyq Ď X which all map to the same element y P Y.

Last time: Recall that the fibers of a map ϕ : X Ñ Y are the sets in ϕ 1 pyq Ď X which all map to the same element y P Y. Last time: Recall that the fibers of a map ϕ : X Ñ Y are the sets in ϕ 1 pyq Ď