StudyGuideDirectProducts

StudyGuideDirectProducts - and n are relatively prime, then...

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

View Full Document Right Arrow Icon
Direct Products Study Guide Outline 1. External Direct Products If G and H are groups, the direct product of G and H , denoted G × H (or G H in the book), is defined as follows: The elements of G × H are ordered pairs ( g,h ), where g G and h H . The operation on G × H is defined componentwise, i.e. ( g 1 ,h 1 )( g 2 ,h 2 ) = ( g 1 g 2 ,h 1 h 2 ) . The order of G × H is the product of the orders of G and H : | G × H | = | G || H | Problems: 10, 11, 12 2. Orders of Elements Let G and H be groups, and let ( g,h ) G × H . Then the order of ( g,h ) is the least common multiple of the orders of g and h : ± ± ( g,h ) ± ± = lcm ( | g | , | h | ) Problems: 13, 14, 15, 16 3. Internal Direct Products Let G be a group, let H and K be subgroups of G , and suppose that 1. H K = { e } , 2. Every element of G can be expressed as the product of an element of H with an element of K , and 3. Elements of H commute with elements of K . Then G H × K . This theorem lets us recognize that a given group is a direct product. For example, V Z 2 × Z 2 and D 6 D 3 × Z 2 More generally, D 2 n D n × Z 2 for odd values of n .
Background image of page 1

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

View Full DocumentRight Arrow Icon
4. Direct Products of Cyclic Groups If m
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: and n are relatively prime, then Z mn ≈ Z m × Z n This makes it possible to write any cyclic group as a direct product of cyclic groups whose orders are prime powers. For example, Z 360 ≈ Z 8 × Z 9 × Z 5 . This is most often used to check whether two direct products of cyclic groups are isomorphic. Problems: 21, 22 5. Structure of U-groups If m and n are relatively prime, then U ( mn ) ≈ U ( m ) × U ( n ) This makes it possible to express any U-group as a direct products of U-groups for prime powers. For example, U (360) ≈ U (8) × U (9) × U (5). If p is prime, then U ( p ) ≈ Z p-1 More generally, if p is an odd prime, then U ( p k ) ≈ Z φ ( p k ) where φ ( n ) = | U ( n ) | . Things work slightly differently for powers of two: U (2 k ) ≈ Z 2 × Z 2 k-2 Problems: 17, 18, 19, 20...
View Full Document

This document was uploaded on 11/03/2010.

Page1 / 2

StudyGuideDirectProducts - and n are relatively prime, then...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online