My Math Forum  

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

Abstract Algebra Abstract Algebra Math Forum


Thanks Tree2Thanks
  • 2 Post By Maschke
Reply
 
LinkBack Thread Tools Display Modes
March 27th, 2015, 10:06 PM   #1
Member
 
Joined: Jun 2012
From: San Antonio, TX

Posts: 84
Thanks: 3

Math Focus: Differential Equations, Mathematical Modeling, and Dynamical Systems
Why does each g in G lie in the coset gH? Where H is a subgroup of G

Textbook: A First Course in Abstract Algebra Rotman

I was following a proof of Lagrange's Theorem that states that if $H$ is a subgroup of a finite group G, then $|H|$ is a divisor of $|G|$.

In the proof it says let $\{ a_1H, a_2H,\ldots, a_tH \}$ be the family of all the distinct cosets of $H$ in $G$. Then
$$G=a_1H \cup a_2H \cup \cdots \cup a_tH$$
because each $g\in G$ lies in the coset $gH$

I omit the rest of the proof because I am having trouble seeing why that is so.

Take, for example, $G=S_3$ and $H=\left< (1 \ 2) \right>$.

$\begin{alignat*}{3} H &= \{ (1), (1 \ 2)\} &&= (1 \ 2)H \\ (1 \ 3)H &= \{ (1 \ 3), (1 \ 2 \ 3) \} &&= (1 \ 2 \ 3)H \\ (2 \ 3)H &= \{ (2 \ 3), (1 \ 3 \ 2) \} &&= (1 \ 3 \ 2)H \end{alignat*}$

Clearly, each $g\in G$ is in $gH$, but why exactly is this so?

EDIT: I seem to gain understanding to my question once I type up a question and post it to this forum. My guess is that it has to do with the identity of H.

Last edited by MadSoulz; March 27th, 2015 at 10:13 PM.
MadSoulz is offline  
 
March 27th, 2015, 11:03 PM   #2
Senior Member
 
Joined: Aug 2012

Posts: 2,306
Thanks: 706

Quote:
Originally Posted by MadSoulz View Post
EDIT: I seem to gain understanding to my question once I type up a question and post it to this forum. My guess is that it has to do with the identity of H.
Right. If $e$ is the identity and $H$ is a subgroup then $e \in H$ therefore $g = ge \in gH$.
Thanks from MadSoulz and Country Boy
Maschke is offline  
Reply

  My Math Forum > College Math Forum > Abstract Algebra

Tags
coset, lie, subgroup



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
subgroup vs. coset goodfeeling Abstract Algebra 3 January 19th, 2013 11:24 PM
Can anyone help me with this coset problem? Elladeas Abstract Algebra 2 May 7th, 2011 08:51 AM
Subgroup/Normal Subgroup/Automorphism Questions envision Abstract Algebra 3 October 4th, 2009 10:37 PM
Subgroup/Normal Subgroup/Factor Group Questions envision Abstract Algebra 1 October 4th, 2009 03:24 AM
member of coset with same order as coset signaldoc Abstract Algebra 3 February 19th, 2009 05:50 PM





Copyright © 2019 My Math Forum. All rights reserved.