ahw5 - Math 4124 Wednesday, March 16 Fifth Homework...

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

View Full Document Right Arrow Icon
Math 4124 Wednesday, March 16 Fifth Homework Solutions 1. Let G be a group of order 10 with a normal subgroup H of order 2. (a) Prove that H is contained in the center of G (hint: If 1 6 = x H and g G , then 1 6 = gxg - 1 H ). (b) Prove that G is abelian. (c) Let a G \ H . Prove that aH has order 5 in G / H . (d) Prove that G has an element y of order 5. (e) Let 1 6 = x H . Prove that the order of xy is neither 2 nor 5. (f) Prove that G is cyclic. (g) Prove that G = Z / 10 Z . (a) Since | H | = 2, it has two elements, one of which is the identity 1; we’ll call the other element x . Let g G . The g 1 g - 1 = 1. Also gxg - 1 H \ 1 (because H ± G ), hence gxg - 1 = x . It follows that g commutes with every element of H and we deduce that H is contained in the center of G . (b) Note that G / H is cyclic because it has order 5, which is a prime, and all groups of prime order are cyclic. Since H is contained in the center of G , we conclude that G is abelian. (c) By Lagrange’s theorem, | aH | divides | G / H | = 5. Since aH 6 = 1 because a / H , we see that | aH | = 5. (d) Since | aH | = 5, we see that | a | 6 = 1 or 2. Also | a | divides 10 by Lagrange’s theorem. Therefore | a | = 5 or 10. If | a | = 5, we’re finished. On the other hand if | a | = 10, then | a 2
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 4124 taught by Professor Staff during the Spring '08 term at Virginia Tech.

Page1 / 3

ahw5 - Math 4124 Wednesday, March 16 Fifth Homework...

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