ahw2 - Math 5125 Monday, September 5 Second Homework...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 5125 Monday, September 5 Second Homework Solutions 1. Let G be a group with distinct normal subgroups A , B such that | G : A | = | G : B | = 2. (a) Prove that AB = G . (b) Prove that A / A B and B / A B are distinct normal subgroups of G / A B of order 2 and index 2. (c) Deduce that G has at least 3 subgroups of index 2. (a) AB is a subgroup of G which strictly contains A . Since A has index 2 in G , it follows that AB has index 1 in G and so AB = G . (b) Note that A B is a normal subgroup of G because A and B are normal subgroups. By the subgroup correspondence theorem, A / A B and B / A B are normal sub- groups of G / A B , and they are distinct because A and B are distinct. Since A and B have index 2 in G , we see that A / A B and B / A B have index 2 in G / A B . Finally A / A B = AB / B = G / B , which shows that A / A B has order 2, and similarly B / A B has order 2....
View Full Document

Page1 / 2

ahw2 - Math 5125 Monday, September 5 Second 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