# LS10-29 - S UMMARY OF 10/29 L ECTURE Before proving the...

This preview shows pages 1–2. Sign up to view the full content.

S UMMARY OF 10/29 L ECTURE Before proving the important result stated at the end of the previous lecture, we started with an easy observation: given a homomorphism φ : G H , we showed that φ maps G surjectively onto its image, i.e. φ : G im φ . We further noted that this map need not be surjective (e.g. the homomorphism φ : Q × Q × which takes α 7→ α 2 does not inject into im φ ; 4 = φ ( ± 2) ). Next, some notation: given a group G and a normal subgroup N , to save space we will write G := G/N . Every element of G looks like a coset, a translation of N , i.e. is of the form gN for some g G ; we will write g := gN . I’ll repeat my mantra: there are elements we can distinguish in G , which can no longer be distinguished when viewed in G . Using our new notation: it’s possible to have g = h for two distinct elements g and h of G . Beware! We are now ready to prove the following important result: Theorem 1. Let φ : G H be a homomorphism. Then ker φ G and G/ ker φ im φ. Proof. We proved last time that ker φ is a normal subgroup of G , so it remains only to prove the isomorphism in the statement of the theorem. The following diagram will be helpful: (1) G π φ / im φ G μ = { { { { where G := G/ ker φ , whose elements we denote by g := g ker φ . Here π is the natural surjection g 7-→ g (why is this a surjection?), φ surjects onto im φ (as discussed above), and μ is the isomorphism we wish to construct. How on earth do we construct such a

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern