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An Aside from Number Theory: The Euler Phi-function (9/20/13)

An Aside from Number Theory: The Euler Phi-function (9/20/13). Definition. The Euler Phi-function of a positive integer n , denoted ( n ) is the number of elements in {1, 2,..., n - 1} which are relatively prime to n. Here’s how to compute ( n ):

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An Aside from Number Theory: The Euler Phi-function (9/20/13)

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  1. An Aside from Number Theory:The Euler Phi-function (9/20/13) • Definition. The Euler Phi-function of a positive integer n, denoted (n) is the number of elements in {1, 2,..., n -1} which are relatively prime to n. • Here’s how to compute (n): • Factor n into its unique prime power factorization (possible by the Fundamental Theorem of Arithmetic). • For each factor of the form pk (where p is prime), (pk) = pk – pk-1. • Multiply the individual answers together to get the final (n). • What is (17)? • What is (20)? • What is (108)? • What is (120)?

  2. More on Subgroups • We saw last time that a rich source of subgroups inside a group G is its cyclic subgroups. But not all subgroups are cyclic. • Q is a subgroup of R. Is Q cyclic? • Find a non-cyclic proper subgroup of D4. Find another. How many are there? • Find a non-cyclic proper subgroup of U(20). • Of course, if G itself is non-cyclic, there’s a non-cyclic subgroup right there.

  3. The Center of a Group • Definition: Let G be a group and let Z(G) = {a G | a x = x a for all x  G}. Z(G) is called the center of G. • Note: This notation is Z, not Z. Both come from German. Z is for Zentrum, which means center. Z is for zahlen, which means to count. • This idea is only of interest in non-abelian groups. (Why?) • What is the center of D4? Of D5? Of Dn(two cases!)? • What is the center of GL(2, R)? Of SL(2, R)? • Theorem. The center of any group G is a subgroup of G. • Proof of theorem?

  4. Assignment for Monday • Continuing in Chapter 3, please do Exercises 11, 15 (Can you generalize this result? What was true of 7 and 3 that would be true of other number pairs?), 18, 19, 21, 22, 25, 26, 27, 28, 30 on pages 69-70.

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