Solved

Abstract Algebra: Direct Products

Posted on 2008-10-29
4
2,186 Views
Last Modified: 2012-08-13
I have two problems that i need help with! I'm trying to study for an exam and cannot figure out these review problems

first problem:
The group S3 (+) Z2 is isomorphic to one of the following groups: Z12,
Z6 (+) Z2, A4, D6. Determine which one by elimination.

I know to be isomorphic denoted phi from a group G to a group G' is a one to one mapping from G onto G' that preserves the group operation. So phi(ab)= phi(a) phi(b). I also know to prove one is isomorphic, need to have mapping (define a function phi from G to G'), need to be 1 to 1 (assume that phi(a)=phi(b) and prove that a=b), needs to be onto (prove for any element g' in G', find an element g in G such that phi(g)=g'), and prove that phi is operation preserving (show phi(ab)= phi(a)phi(b)).

I also know that S3 (+) Z2 the first group S3 is multiplicative and second group Z2 is additive so dealing with mixed binary operations.
I think S3 (+) Z2 = {(1), 0), ((1 2), 0), ((1 2 3), 1)}
then ((1 2, 0) * ((1 2 3), 1)= ((1 2)(1 2 3), 1)= ((1)(2 3 1), 1). But i'm not too sure.

second problem:
Find a subgroup of Z12 (+) Z4 (+) Z15 that is of order 9.

I know that the direct product is the set of all n-tuples for which the ith componenet is an element of Gi and the operation is componentwise.

I'm confused how to find direct products, can someone help me? Thanks so much, greatly appreciated!
0
Comment
Question by:dongyowlin
[X]
Welcome to Experts Exchange

Add your voice to the tech community where 5M+ people just like you are talking about what matters.

  • Help others & share knowledge
  • Earn cash & points
  • Learn & ask questions
  • 2
4 Comments
 
LVL 84

Accepted Solution

by:
ozo earned 500 total points
ID: 22835641
A4

Z3 (+) Z1 (+) Z3
0
 

Author Comment

by:dongyowlin
ID: 22835723
how does A4 equal to Z3 (+) Z1 (+) Z3?
A i know means alternating group. but im still confused...

0
 
LVL 84

Expert Comment

by:ozo
ID: 22838057
A4 is isomorphic to  S3 (+) Z2

Z3 (+) Z1 (+) Z3 is a subgroup of Z12 (+) Z4 (+) Z15 that is of order 9.
0
 
LVL 20

Expert Comment

by:thehagman
ID: 23106387
Unfortunately, all candidates have the correct order (12), so this cannot be used for elimination.
Since S3 is not abelian, the direct product isn't abelian either.
This eliminates Z12 and Z2(+)Z6.
The only subgroup of order 3 in D6 consists of the rotations with multiples of 120°.
Such a rotation never commutes with any element of order 2 (i.e. reflection); this eliminates D6.

2)
Z12 and Z15 have a subgroup isomorphic to Z3 each, together they make a group of order 9.
I.e., the subgroup generated by (4,0,0) and (0,0,5) is what you are looking for.
0

Featured Post

Technology Partners: We Want Your Opinion!

We value your feedback.

Take our survey and automatically be enter to win anyone of the following:
Yeti Cooler, Amazon eGift Card, and Movie eGift Card!

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Foreword (May 2015) This web page has appeared at Google.  It's definitely worth considering! https://www.google.com/about/careers/students/guide-to-technical-development.html How to Know You are Making a Difference at EE In August, 2013, one …
This article provides a brief introduction to tissue engineering, the process by which organs can be grown artificially. It covers the problems with organ transplants, the tissue engineering process, and the current successes and problems of the tec…
Although Jacob Bernoulli (1654-1705) has been credited as the creator of "Binomial Distribution Table", Gottfried Leibniz (1646-1716) did his dissertation on the subject in 1666; Leibniz you may recall is the co-inventor of "Calculus" and beat Isaac…
Finds all prime numbers in a range requested and places them in a public primes() array. I've demostrated a template size of 30 (2 * 3 * 5) but larger templates can be built such 210  (2 * 3 * 5 * 7) or 2310  (2 * 3 * 5 * 7 * 11). The larger templa…

726 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question