1 / 4

Centralizer of an Element (9/23 )

Centralizer of an Element (9/23 ). Definition. If G is a group and a  G , the Centralizer of a , denoted C( a ), is { g  G | a g = g a }. That is, it is the set of all elements of G which commute with the fixed element a . Determine the centralizers of all elements of D 4 .

naava
Download Presentation

Centralizer of an Element (9/23 )

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Centralizer of an Element (9/23) • Definition. If G is a group and a G,theCentralizer of a, denoted C(a), is {g  G| a g = g a}. That is, it is the set of all elements of G which commute with the fixed element a. • Determine the centralizers of all elements of D4. • C(a) is always a subgroup of G. (Proof?) • What is the relationship between C(a) and Z(G)?

  2. Another Number Theory Aside: The Extended Euclidean Algorithm • Theorem. If d is the greatest common divisor of positive integers a and b (notation: d = GCD(a, b)), then there exist integers x and y such that d = x a + y b. • That is, the GCD of two positive integers can be written as a “linear combination” of those integers. • Write GCD(3, 7) in this form. • Write GCD(6,10) in this form. • Write GCD(10457365,54876) in this form. (??) • There is a fast algorithm to do this.

  3. Smallest Subgroup Containing... • Definition. If a and b are elements of a group G, by the notation a, b, we mean the smallest subgroup of G which contains both a and b. This is called the subgroup generated by a and b. Similarly a, b, c, etc. • In Z, what is2, 3? What is 4, 6? What is n, m? • In Z10, what is 3, 5? • In D4, what is H, V? What is H, D?

  4. Assignment for Wednesday • We finish up Chapter 3. Do Exercises 33, 35, 38, 40, 41, 43. 44, 45, 53 and 55 on pages 71-73.

More Related