Last Time: Isomorphism Theorems

Home , Isomorphism theorems

2. Let A, B ď G and assume A ď NGpBq. 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.

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. 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.

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 NGpBq. Then AB{B – A{pA X Bq (with appropriate statements about normality). 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.

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 NGpBq. 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. 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.

Φ: G Ñ G{K deﬁned by g ÞÑ gK, except split into two parts by

π ϕ G G{H G{K

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. 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

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). 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 if and only if ϕpgNq “ ϕpg1Nq for all g1 P gN if and only if Φpgq “ Φpg1q for all g1N P gN if and only if N ď kerpΦq.

, 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

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: π G G{N ϕ Φ H

General principle: homomorphisms on quotient groups Any map ϕ : G{N Ñ H is a map on cosets gN if and only if ϕpgNq “ ϕpg1Nq for all g1 P gN if and only if Φpgq “ Φpg1q for all g1N P gN if and only if N ď kerpΦq.

Essentially, ϕ pulls back to a map Φ: G Ñ H where Φpgq “ ϕpgNq. Then ϕ is well-deﬁned

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. if and only if ϕpgNq “ ϕpg1Nq for all g1 P gN if and only if Φpgq “ Φpg1q for all g1N P gN if and only if N ď kerpΦq.

Then ϕ is well-deﬁned

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 if and only if ϕpgNq “ ϕpg1Nq for all g1 P gN if and only if Φpgq “ Φpg1q for all g1N P gN if and only if 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. 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 if and only if ϕpgNq “ ϕpg1Nq for all g1 P gN if and only if Φpgq “ Φpg1q for all g1N P gN if and only if 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: π G G{N ϕ Φ H For all N ď A, B ď G, this bijection additionally satisfies 1. A Ĳ G if and only if A¯ Ĳ G¯, 2. A ď B if and only if A¯ ď B¯, 3. if A ď B, then |B : A| “ |B¯ : A¯|, 4. xA, By “ xA,¯ B¯y.

Fourth (lattice) isomorphism theorem

Theorem Let N Ĳ G. There natural projection π : G Ñ G{N gives a bijection

tA | N ď A ď Gu ÐÑ tA¯ | A¯ ď G{Nu

where A¯ “ πpAq “ A{N. Fourth (lattice) isomorphism theorem

Theorem Let N Ĳ G. There natural projection π : G Ñ G{N gives a bijection

tA | N ď A ď Gu ÐÑ tA¯ | A¯ ď G{Nu

where A¯ “ πpAq “ A{N.

For all N ď A, B ď G, this bijection additionally satisﬁes 1. A Ĳ G if and only if A¯ Ĳ G¯, 2. A ď B if and only if A¯ ď B¯, 3. if A ď B, then |B : A| “ |B¯ : A¯|, 4. xA, By “ xA,¯ B¯y. Lattice of D16

t1, r, r2, r3, r4, r5, r6, r7, s, sr, sr2, sr3, sr4, sr5, sr6, sr7u

t1, r2, r4, r6, s, sr2, sr4, sr6u t1, r, r2, r3, r4, r5, r6, r7u t1, r2, r4, r6, sr, sr3, sr5, sr7u

t1, r4, sr2, sr6u t1, r4, s, sr4u t1, r2, r4, r6u t1, r4, sr3, sr7u t1, r4, sr, sr5u

t1, sr6u t1, sr2u t1, sr4u t1, su t1, r4u t1, sr3u t1, sr7u t1, sr5u t1, sru

t1u Lattice of D16

t1, r, r2, r3, r4, r5, r6, r7, s, sr, sr2, sr3, sr4, sr5, sr6, sr7u

t1, r2, r4, r6, s, sr2, sr4, sr6u t1, r, r2, r3, r4, r5, r6, r7u t1, r2, r4, r6, sr, sr3, sr5, sr7u

t1, r4, sr2, sr6u t1, r4, s, sr4u t1, r2, r4, r6u t1, r4, sr3, sr7u t1, r4, sr, sr5u

t1, sr6u t1, sr2u t1, sr4u t1, su t1, r4u t1, sr3u t1, sr7u t1, sr5u t1, sru

t1u Lattice of D16

t1, r, r2, r3, r4, r5, r6, r7, s, sr, sr2, sr3, sr4, sr5, sr6, sr7u

t1, r2, r4, r6, s, sr2, sr4, sr6u t1, r, r2, r3, r4, r5, r6, r7u t1, r2, r4, r6, sr, sr3, sr5, sr7u

t1, r4, sr2, sr6u t1, r4, s, sr4u t1, r2, r4, r6u t1, r4, sr3, sr7u t1, r4, sr, sr5u

t1, sr6u t1, sr2u t1, sr4u t1, su t1, r4u t1, sr3u t1, sr7u t1, sr5u t1, sru

t1u t1, sr6u t1, sr2u t1, sr4u t1, su t1, sr3u t1, sr7u t1, sr5u t1, sru

t1u

Lattice of D16

tZ, rZ, r2Z, r3Z, sZ, srZ, sr2Z, sr3Zu

tZ, r2Z, sZ, sr2Zu tZ, rZ, r2Z, r3Zu tZ, r2Z, srZ, sr3Zu

tZ, sr2Zu tZ, sZu tZ, r2Zu tZ, sr3Zu tZ, srZu

Z X

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. Hölderprogram: 1. Classify all finite simple groups. 2. Find all ways of “putting simple groups together”.

The H¨olderprogram, a story Goal: Classify, up to isomorphism, all ﬁnite groups. X

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. Hölderprogram: 1. Classify all finite simple groups. 2. Find all ways of “putting simple groups together”.

The H¨olderprogram, 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” X

“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. H¨olderprogram: 1. Classify all ﬁnite simple groups. 2. Find all ways of “putting simple groups together”.

Base case: We say a group G is simple if |G| ą 1 and the only normal subgroups of G are G and 1. H¨olderprogram: 1. Classify all ﬁnite simple groups. 2. Find all ways of “putting simple groups together”.

H¨olderprogram: 1. Classify all ﬁnite simple groups. 2. Find all ways of “putting simple groups together”.

The Hölderprogram, 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. Hölderprogram: 1. Classify all finite simple groups. 2. Find all ways of “putting simple groups together”. The Hölderprogram, 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. Hölderprogram: 1. Classify all finite simple groups. X 2. Find all ways of “putting simple groups together”. For example, A “ tZp | p primeu is one of the 18 infinite families of finite simple groups. Namely, if 1 p is prime, then Zp is simple (by Lagrange); and if p ‰ p , then Zp fl Zp1 . The sort of theorem that went into the classification is as follows: Theorem (Feit-Thompson). If G is simple and of odd order, then G – Zp for some prime p.

(The proof of this theorem took about 255 pages of sophisticated and technical mathematics. We won’t do it.)

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. Namely, if 1 p is prime, then Zp is simple (by Lagrange); and if p ‰ p , then Zp fl Zp1 . The sort of theorem that went into the classification is as follows: Theorem (Feit-Thompson). If G is simple and of odd order, then G – Zp for some prime p.

(The proof of this theorem took about 255 pages of sophisticated and technical mathematics. We won’t do it.)

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 “ tZp | p primeu is one of the 18 infinite families of finite simple groups. The sort of theorem that went into the classification is as follows: Theorem (Feit-Thompson). If G is simple and of odd order, then G – Zp for some prime p.

(The proof of this theorem took about 255 pages of sophisticated and technical mathematics. We won’t do it.)

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 “ tZp | p primeu is one of the 18 infinite families of finite simple groups. Namely, if 1 p is prime, then Zp is simple (by Lagrange); and if p ‰ p , then Zp fl Zp1 . 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 “ tZp | p primeu is one of the 18 infinite families of finite simple groups. Namely, if 1 p is prime, then Zp is simple (by Lagrange); and if p ‰ p , then Zp fl Zp1 . The sort of theorem that went into the classification is as follows: Theorem (Feit-Thompson). If G is simple and of odd order, then G – Zp for some prime p.

(The proof of this theorem took about 255 pages of sophisticated and technical mathematics. We won’t do it.) Namely, SLnpFq “ tM P GLnpFq | detpMq “ 1u is ﬁnite if F is ﬁnite. But SLnpFq is not simple, since ZpGq is always normal in G, and λ1 λ2 ZpSLnpFqq “ ¨ . ˛ λi P F, λ1λ2 ¨ ¨ ¨ λn “ 1 ‰ 1 $ .. ˇ , ˚ ‹ ˇ &’ ˚ λ ‹ ˇ ./ ˚ n‹ ˝ ‚ ˇ if |F| ą 1. It turns out, though, thatˇ SLnpFq{ZpSLnpFqq is simple. %’ ˇ -/ And the groups in B are distinct forˇ distinct n and F.

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 “ tSLnpFq{ZpSLnpFqq | n P Zě2, F is a finite fieldu. But SLnpFq is not simple, since ZpGq is always normal in G, and λ1 λ2 ZpSLnpFqq “ ¨ . ˛ λi P F, λ1λ2 ¨ ¨ ¨ λn “ 1 ‰ 1 $ .. ˇ , ˚ ‹ ˇ &’ ˚ λ ‹ ˇ ./ ˚ n‹ ˝ ‚ ˇ if |F| ą 1. It turns out, though, thatˇ SLnpFq{ZpSLnpFqq is simple. %’ ˇ -/ And the groups in B are distinct forˇ distinct n and F.

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 “ tSLnpFq{ZpSLnpFqq | n P Zě2, F is a finite fieldu. Namely, SLnpFq “ tM P GLnpFq | detpMq “ 1u is finite if F is finite. But SLnpFq is not simple, since ZpGq is always normal in G, and λ1 λ2 ZpSLnpFqq “ ¨ . ˛ λi P F, λ1λ2 ¨ ¨ ¨ λn “ 1 ‰ 1 $ .. ˇ , ˚ ‹ ˇ &’ ˚ λ ‹ ˇ ./ ˚ n‹ ˇ if | | ą 1. ˝ ‚ ˇ F ’ ˇ / % ˇ - And the groups in B are distinct for distinct n and F.

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 “ tSLnpFq{ZpSLnpFqq | n P Zě2, F is a finite fieldu. Namely, SLnpFq “ tM P GLnpFq | detpMq “ 1u is finite if F is finite. But SLnpFq is not simple, since ZpGq is always normal in G, and λ1 λ2 ZpSLnpFqq “ ¨ . ˛ λi P F, λ1λ2 ¨ ¨ ¨ λn “ 1 ‰ 1 $ .. ˇ , ˚ ‹ ˇ &’ ˚ λ ‹ ˇ ./ ˚ n‹ ˇ if | | ą 1. It turns˝ out, though,‚ thatˇ SL p q{ZpSL p qq is simple. F ’ ˇ n F n F / % ˇ - 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 “ tSLnpFq{ZpSLnpFqq | n P Zě2, F is a finite fieldu. Namely, SLnpFq “ tM P GLnpFq | detpMq “ 1u is finite if F is finite. But SLnpFq is not simple, since ZpGq is always normal in G, and λ1 λ2 ZpSLnpFqq “ ¨ . ˛ λi P F, λ1λ2 ¨ ¨ ¨ λn “ 1 ‰ 1 $ .. ˇ , ˚ ‹ ˇ &’ ˚ λ ‹ ˇ ./ ˚ n‹ ˝ ‚ ˇ if |F| ą 1. It turns out, though, thatˇ SLnpFq{ZpSLnpFqq is simple. %’ ˇ -/ And the groups in B are distinct forˇ distinct n and F. Q8 D8 2 2 xiy xjy xky xs, r y xry xsr, r y

x´1y xr2y

Example: B – 1 ˆ B Ĳ A ˆ B and pA ˆ Bq{p1 ˆ Bq – A. Example: The lattices for Q8 and D8 are Q8 D8 2 2 xiy xjy xky xs, r y xry xsr, r y