# Lecture 5 - 5 Basic Properties of Groups Lemma 5.1 Let G be...

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

5. Basic Properties of Groups Lemma 5.1. Let G be a group. (1) G contains exactly one identity element. (2) Every element of G contains exactly one inverse. (3) Let a and b be any two elements of G . Then the equation ax = b has exactly one solution in G , namely x = a - 1 b . (4) Let a and b be any two elements of G . Then the equation ya = b has exactly one solution, namely y = ba - 1 . (5) For every a G , ( a - 1 ) - 1 = a. In words the inverse of the inverse of a is a . (6) For every a and b in G , ( ab ) - 1 = b - 1 a - 1 . That is, the inverse of a product is the product of the inverses, in the opposite order. Proof. We first prove (1). By definition G has to contain at least one identity element. Suppose that both e and f are identity elements in G . We compute the product ef . As e is an identity in G , ef = f. On the other hand as f is an identity in G , ef = e. Thus e = ef = f . Thus the identity is unique. Hence (1). Now we prove (2). Suppose that g is an element of G . Then g has at least one inverse by definition. Suppose that there were two elements h and k that were both inverses of g . We compute hgk (by associativity we can drop the parentheses). On the one hand we get

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