2. Let A; B ¤ G and assume A ¤ NGpBq. Then AB{B – A{pA X Bq (with appropriate statements about normality). 3. Let A; B IJ G with A ¤ 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 ' : G Ñ H is a homomorphism of groups, then kerp'q IJ G and G{kerp'q – 'pGq: 3. Let A; B IJ G with A ¤ 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 ' : G Ñ H is a homomorphism of groups, then kerp'q IJ G and G{kerp'q – 'pGq: 2. Let A; B ¤ G and assume A ¤ 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 ' : G Ñ H is a homomorphism of groups, then kerp'q IJ G and G{kerp'q – 'pGq: 2. Let A; B ¤ G and assume A ¤ NGpBq. Then AB{B – A{pA X Bq (with appropriate statements about normality). 3. Let A; B IJ G with A ¤ 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 ' : G Ñ H is a homomorphism of groups, then kerp'q IJ G and G{kerp'q – 'pGq: 2. Let A; B ¤ G and assume A ¤ NGpBq. Then AB{B – A{pA X Bq (with appropriate statements about normality). 3. Let A; B IJ G with A ¤ 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. The first step was to show this was well-defined (independent of choice of representative in gH). Another way: ' is basically the map Φ: G Ñ G{K defined by g ÞÑ gK; except split into two parts by π ' G G{H G{K Φ 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 ' : G{H Ñ G{K defined by gH ÞÑ gK: Another way: ' is basically the map Φ: G Ñ G{K defined by g ÞÑ gK; except split into two parts by π ' G G{H G{K Φ 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 ' : 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 ' : 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: ' is basically the map Φ: G Ñ G{K defined by g ÞÑ gK; except split into two parts by π ' G G{H G{K Φ if and only if 'pgNq “ 'pg1Nq for all g1 P gN if and only if Φpgq “ Φpg1q for all g1N P gN if and only if N ¤ kerpΦq. , so the image is determined by g itself. Essentially, ' pulls back to a map Φ: G Ñ H where Φpgq “ 'pgNq: Then ' is well-defined So to define a homomorphism ' : G{N Ñ H, you can instead define a homomorphism Φ: G Ñ H and check that N ¤ kerpΦq. In this case, we say Φ factors through N and ' 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 H General principle: homomorphisms on quotient groups Any map ' : G{N Ñ H is a map on cosets gN if and only if 'pgNq “ 'pg1Nq for all g1 P gN if and only if Φpgq “ Φpg1q for all g1N P gN if and only if N ¤ kerpΦq. Essentially, ' pulls back to a map Φ: G Ñ H where Φpgq “ 'pgNq: Then ' is well-defined So to define a homomorphism ' : G{N Ñ H, you can instead define a homomorphism Φ: G Ñ H and check that N ¤ kerpΦq. In this case, we say Φ factors through N and ' 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 H General principle: homomorphisms on quotient groups Any map ' : G{N Ñ H is a map on cosets gN, so the image is determined by g itself. if and only if 'pgNq “ 'pg1Nq for all g1 P gN if and only if Φpgq “ Φpg1q for all g1N P gN if and only if N ¤ kerpΦq. Then ' is well-defined So to define a homomorphism ' : G{N Ñ H, you can instead define a homomorphism Φ: G Ñ H and check that N ¤ kerpΦq. In this case, we say Φ factors through N and ' 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 H General principle: homomorphisms on quotient groups Any map ' : G{N Ñ H is a map on cosets gN, so the image is determined by g itself. Essentially, ' pulls back to a map Φ: G Ñ H where Φpgq “ 'pgNq: if and only if 'pgNq “ 'pg1Nq for all g1 P gN if and only if Φpgq “ Φpg1q for all g1N P gN if and only if N ¤ kerpΦq. So to define a homomorphism ' : G{N Ñ H, you can instead define a homomorphism Φ: G Ñ H and check that N ¤ kerpΦq. In this case, we say Φ factors through N and ' 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 H General principle: homomorphisms on quotient groups Any map ' : G{N Ñ H is a map on cosets gN, so the image is determined by g itself. Essentially, ' pulls back to a map Φ: G Ñ H where Φpgq “ 'pgNq: Then ' is well-defined if and only if Φpgq “ Φpg1q for all g1N P gN if and only if N ¤ kerpΦq. So to define a homomorphism ' : G{N Ñ H, you can instead define a homomorphism Φ: G Ñ H and check that N ¤ kerpΦq. In this case, we say Φ factors through N and ' 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 H General principle: homomorphisms on quotient groups Any map ' : G{N Ñ H is a map on cosets gN, so the image is determined by g itself. Essentially, ' pulls back to a map Φ: G Ñ H where Φpgq “ 'pgNq: Then ' is well-defined if and only if 'pgNq “ 'pg1Nq for all g1 P gN if and only if N ¤ kerpΦq. So to define a homomorphism ' : G{N Ñ H, you can instead define a homomorphism Φ: G Ñ H and check that N ¤ kerpΦq. In this case, we say Φ factors through N and ' 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 H General principle: homomorphisms on quotient groups Any map ' : G{N Ñ H is a map on cosets gN, so the image is determined by g itself. Essentially, ' pulls back to a map Φ: G Ñ H where Φpgq “ 'pgNq: Then ' is well-defined if and only if 'pgNq “ 'pg1Nq for all g1 P gN if and only if Φpgq “ Φpg1q for all g1N P gN So to define a homomorphism ' : G{N Ñ H, you can instead define a homomorphism Φ: G Ñ H and check that N ¤ kerpΦq. In this case, we say Φ factors through N and ' 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 H General principle: homomorphisms on quotient groups Any map ' : G{N Ñ H is a map on cosets gN, so the image is determined by g itself. Essentially, ' pulls back to a map Φ: G Ñ H where Φpgq “ 'pgNq: Then ' is well-defined if and only if 'pgNq “ 'pg1Nq for all g1 P gN if and only if Φpgq “ Φpg1q for all g1N P gN if and only if N ¤ kerpΦq.