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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 .

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Centralizer of an Element (9/23 )

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  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.

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