Ahw6 - Math 3124 Thursday October 20 Sixth Homework Solutions 1 Problem 17.9 Assume that G is a group with a subgroup H such that |H | = 6[G H > 4

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

View Full Document Right Arrow Icon
Math 3124 Thursday, October 20 Sixth Homework Solutions 1. Problem 17.9. Assume that G is a group with a subgroup H such that | H | = 6, [ G : H ] > 4, and | G | < 50. What are the possibilities for | G | ? By Lagrange’s theorem and using the given hypotheses, | G | = 6 [ G : H ] < 50 and [ G : H ] > 4. Therefore the possibilities for [ G : H ] are 5,6,7,8 and hence the possibilities for | G | are 30, 36, 42 and 48. 2. Problem 17.17. Find all the subgroups of Z 36 . Also construct the subgroup lattice. Since Z 36 is a cyclic group of order 36, there will be exactly one subgroup for each positive integer dividing 36. The integers dividing 36 are 2 a 3 b where a , b = 0 , 1 , 2, so there will be 3 * 3 = 9 subgroups. The subgroup lattice will look like ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ u h [ 0 ] i u h [ 12 ] i u h [ 18 ] i u h [ 6 ] i u h [ 2 ] i u h [ 9 ] i u h [ 4 ] i u h [ 3 ] i u h [ 1 ] i 3. From ae = c , we see that neither a nor e is the identity, and from
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
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 3124 taught by Professor Parry during the Fall '08 term at Virginia Tech.

Page1 / 3

Ahw6 - Math 3124 Thursday October 20 Sixth Homework Solutions 1 Problem 17.9 Assume that G is a group with a subgroup H such that |H | = 6[G H > 4

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