My Math Forum  

Go Back   My Math Forum > College Math Forum > Abstract Algebra

Abstract Algebra Abstract Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
October 3rd, 2009, 07:35 PM   #1
Newbie
 
Joined: Oct 2009

Posts: 14
Thanks: 0

Subgroup/Normal Subgroup/Automorphism Questions

In this problem, assume (important!) that G is an Abelian. Set H = {g in G : g^5 = e}. (Warning: Expressions such as x^1/5 are not well defined. Do not use them!)

(a) Show H is a subgroup of G.
(b) Show H is a normal subgroup of G.
(c) Assume further that G is finite and that H = {e}. Show that
the map phi : G --> G given by phi(g) = g^5 is an automorphism.

(Definition: An automorphism is an isomorphism from a group to itself.)
envision is offline  
 
October 4th, 2009, 04:12 AM   #2
Senior Member
 
Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: Subgroup/Normal Subgroup/Automorphism Questions

Quote:
Originally Posted by envision
In this problem, assume (important!) that G is an Abelian. Set H = {g in G : g^5 = e}. (Warning: Expressions such as x^1/5 are not well defined. Do not use them!)

(a) Show H is a subgroup of G.
Use a subgroup test.

Quote:
(b) Show H is a normal subgroup of G.
You need to show that Hg = gH for all g in G. This comes almost directly from the Abelian-ness of G.
Quote:
(c) Assume further that G is finite and that H = {e}. Show that
the map phi : G --> G given by phi(g) = g^5 is an automorphism.
Show that it is operation preserving. This gives you a homomorphism. Have you learned any properties of homomorphism yet? If so, there's one about the kernel of phi. This shows it's a bijection.
If not, show it is injective: show that a?b -> phi(a)?phi(b). An injective function from A->A must be bijective for any finite set A.


Cheers.
cknapp is offline  
October 4th, 2009, 09:14 PM   #3
Newbie
 
Joined: Oct 2009

Posts: 14
Thanks: 0

Re: Subgroup/Normal Subgroup/Automorphism Questions

Quote:
Originally Posted by cknapp
Quote:
Originally Posted by envision
In this problem, assume (important!) that G is an Abelian. Set H = {g in G : g^5 = e}. (Warning: Expressions such as x^1/5 are not well defined. Do not use them!)

(a) Show H is a subgroup of G.
Use a subgroup test.

Quote:
(b) Show H is a normal subgroup of G.
You need to show that Hg = gH for all g in G. This comes almost directly from the Abelian-ness of G.
[quote:16v4wr01]
(c) Assume further that G is finite and that H = {e}. Show that
the map phi : G --> G given by phi(g) = g^5 is an automorphism.
Show that it is operation preserving. This gives you a homomorphism. Have you learned any properties of homomorphism yet? If so, there's one about the kernel of phi. This shows it's a bijection.
If not, show it is injective: show that a?b -> phi(a)?phi(b). An injective function from A->A must be bijective for any finite set A.


Cheers.[/quote:16v4wr01]

concerning part c...
we've learned about homomorphisms. isn't the kernel of phi = e or something like that? i don't really know how to even begin showing part c.
envision is offline  
October 4th, 2009, 11:37 PM   #4
Senior Member
 
Joined: Oct 2007
From: Chicago

Posts: 1,701
Thanks: 3

Re: Subgroup/Normal Subgroup/Automorphism Questions

The kernel is the set of elements which map to e.
You should have learned that a homomorphism is n-1: every element in the range has the same number (n) of elements map to it.
If you can show that only one element maps to e, then you have a 1-1, homomorphism. And since the sets are finite and the same size, this is a bijective homomorphism-- an isomorphism.
cknapp is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
questions, subgroup or automorphism, subgroup or normal



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
normal subgroup tinynerdi Abstract Algebra 5 March 4th, 2010 05:14 AM
Normal subgroup ragnar Abstract Algebra 1 January 2nd, 2010 06:53 AM
Normal Subgroup HairOnABiscuit Abstract Algebra 1 November 25th, 2009 10:46 AM
Subgroup/Normal Subgroup/Factor Group Questions envision Abstract Algebra 1 October 4th, 2009 04:24 AM
normal subgroup fakie6623 Abstract Algebra 3 October 1st, 2007 05:24 PM





Copyright © 2019 My Math Forum. All rights reserved.