aug24 - Math 5125 Wednesday, August 24 August 24, Ungraded...

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

View Full Document Right Arrow Icon
Math 5125 Wednesday, August 24 August 24, Ungraded Homework Exercise 3.3.3 on page 101 Prove that if H is a normal subgroup of G of prime index p , then for all K G either (i) K H or (ii) G = HK and | K : K H | = p . We know that HK G , because H , K G and one of them is normal. Obviously HK H . Also | G / H | = p and p is prime, so by Lagrange’s theorem G / H has only two subgroups, namely H / H and G / H . By subgroup correspondence theorem, we now see that HK = H or G . If K is not contained in H , then we cannot have HK = H and we deduce that HK = G . By the second isomorphism theorem we have HK / H = K / K H , consequently | G / H | = | K / K H | and the result follows. Let H ± G be groups such that G / H = Z / 3 Z × Z / 3 Z . Prove that G has at least four normal subgroups of index 3. Let H ± G such that G / H = Z 3 × Z 3 . We need to prove that G has at least 4 normal sub- groups of index 3. It is easily checked that Z 3 × Z 3 has 4 subgroups of order 3. Since
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 5125 taught by Professor Palinnell during the Fall '07 term at Virginia Tech.

Page1 / 2

aug24 - Math 5125 Wednesday, August 24 August 24, Ungraded...

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