Interview
Mathematics, Abstract algebra
- Course
- Institution
- Book
Interview study book A Book of Abstract Algebra of Charles C Pinter - ISBN: 9780486474175 (Study Notes)
[Show more]
Connected book
Book Title:
Author(s):
- Edition:
- ISBN:
- Edition:
More summaries for
-
SOLUTION MANUAL First Course in Abstract Algebra A 8th Edition by John B. Fraleigh All Chapters Full Complete
-
Intro to Group Theory: Chapter 8 (Finite Permutations) + Exam 1 Review
-
Intro to Group Theory: Chapter 10 (Order of Group Elements)
Seller
Follow
waseemmirza2262
Content preview
Subject : MATHEMATICSPaper 1 : ABSTRACT ALGEBRA
Chapter 1 : Direct Product of Groups
Module 2 : Internal direct product of groups
Anjan Kumar Bhuniya
Department of Mathematics
Visva-Bharati; Santiniketan
West Bengal
1
, Internal direct product of groups
Learning outcomes: 1. Internal direct product of groups.
2. Necessary and sufficient condition for a group to be an
internal direct product.
3. Isomorphism between external and internal direct products.
In the previous module we introduced and characterized external direct products of groups,
that provides us a formulation to think a family of distinct groups as subgroups of a larger group.
To be specific, consider two groups G1 and G2 having the identity elements e1 and e2 respectively.
Then N1 = G1 × {e2 } ' G1 and N2 = {e1 } × G2 ' G2 are two normal subgroups of G1 × G2 . In
this module we consider the reverse problem, that is, given a group G whether there is a family of
subgroups H1 , H2 , · · · , Hk of G such that G ' H1 × H2 × · · · × Hk .
As we can expect, it is not possible for every group in general. Even if it is possible for a group
G, then also the subgroups of which external direct product is isomorphic to G need to satisfy some
conditions. Following result gives us an idea on the conditions that the subgroups need to satisfy.
Henceforth we use simply multiplicative notation instead of ∗ to mean the group operation of the
direct product.
Theorem 0.1. Let G1 , G2 , · · · , Gn be a family of groups. Denote G = G1 × G2 × · · · × Gn and
Hi = {(e1 , · · · , ei−1 , ai , ei+1 , · · · , en ) | ai ∈ Gi }
for every i = 1, 2, · · · , n. Then
1. Hi is a normal subgroup of G and Hi ' Gi for every i = 1, 2, · · · , n;
2. every element of G can be expressed uniquely as h1 h2 · · · hn where hi ∈ Hi for every i =
1, 2, · · · , n.
Proof. 1. First note that (e1 , e2 , · · · , en ) ∈ Hi which ensures that Hi 6= ∅.
2
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller waseemmirza2262. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.49. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
52510 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now