# test1 - Math 4124 Monday February 21 First Test Answer All...

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

Math 4124 Monday, February 21 First Test. Answer All Problems. Please Give Explanations For Your Answers 1. Prove that Q 8 × S 3 is not isomorphic to D 48 . ( Q 8 denotes the quaternion group of order 8 and D 48 denotes the dihedral group of order 48.) (12 points) 2. Let G be a group. Prove that the formula ( g , h ) · x = gxh - 1 for g , h , x G , deﬁnes an action of G × G on G . Show further that if Z ( G ) ± = 1, then there exists a nonidentity element which acts trivially on G (i.e. there exists 1 ± = k G × G such that k · x = x for all x G ). (12 points) 3. Prove that if A and B are subsets of the group G with A B , then C G ( B ) ± C G ( A ) . Is it always true that N G ( B ) ± N G ( A ) ? Justify your answer. (C G and N G denote centralizer and normalizer) (12 points) 4. Let D 16 = ² r , s | r 8 = 1 , s 2 = 1 , rs = sr - 1 ³ , the dihedral group of order 16. Determine for which positive integers n there is a subgroup of order n in D 16 (so you will need to give an example of a subgroup of order

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.

## This note was uploaded on 01/02/2012 for the course MATH 4124 taught by Professor Staff during the Spring '08 term at Virginia Tech.

### Page1 / 2

test1 - Math 4124 Monday February 21 First Test Answer All...

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

View Full Document
Ask a homework question - tutors are online