/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 For any group \(G\) prove that \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 8

For any group \(G\) prove that \(G\) is abelian if and only if \((a b)^{2}=a^{2} b^{2}\) for all \(a, b \in G\).

Short Answer

Expert verified
Yes, a group \(G\) is abelian if and only if \((ab)^2 = a^2b^2\) for all \(a, b \in G\). The first part of the proof uses the fact that in an abelian group the order of the elements doesn't matter, while the second part used the provided equation to prove that \(G\) must be an abelian group.

Step by step solution

01

Assume G is an abelian group

Assume that \(G\) is an abelian group. According to the definition of an abelian group, this means that for all \(a, b \in G\), we have \(ab = ba\).
02

Proof for \((ab)^2 = a^2b^2\)

We use the definition of an abelian group to show that \( (ab)^2 = a^2b^2\). Square both sides of the commutation relationship to get \((ab)^2 = (ba)^2\), then apply the associative property to get \(a^2b^2\). Thus, we have shown that if \(G\) is abelian, then \((ab)^2 = a^2b^2\).
03

Assuming \((ab)^2 = a^2b^2\) for all \(a, b \ in G\)

Next, we assume that \((ab)^2 = a^2b^2\) holds for all \(a, b \ in G\). We want to show that this condition will imply \(G\) is an abelian group.
04

Show that \(ab = ba\) for all \(a, b \in G\)

We show that with \((ab)^2 = a^2b^2\) we can derive \(ab = ba\). Start with \(a^2b^2 = (ab)^2 = abab\). Therefore, \(a^{-1}a^2b^2b^{-1} = a^{-1}ababb^{-1}\), which gives us \(ab = ba\). This concludes that \( G\) is an abelian group. Therefore, we have shown that \((ab)^2 = a^2b^2\) if and only if \(G\) is an abelian group.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

For each of the following encoding functions, find the minimum distance between the code words. Discuss the errordetecting and error-correcting capabilities of each code. a) \(\begin{array}{rl}E: \mathbf{Z}_{2}^{2} \rightarrow \mathbf{Z}_{2}^{5} \\\ 00 \rightarrow 00001 & 01 \rightarrow 01010 \\ 10 \rightarrow 10100 & 11 \rightarrow 11111\end{array}\) b) \(E: \mathbf{Z}_{2}^{2} \rightarrow \mathbf{Z}_{2}^{10}\) \(\begin{array}{cl}00 \rightarrow 0000000000 & 01 \rightarrow 0000011111 \\ 10 \rightarrow 1111100000 & 11 \rightarrow 1111111111\end{array}\) \begin{array}{ll} \text { c) } E: \mathbf{Z}_{2}^{3} \rightarrow \mathbf{Z}_{2}^{6} & \\ 000 \rightarrow 000111 & 001 \rightarrow 001001 \\ 010 \rightarrow 010010 & 011 \rightarrow 011100 \\ 100 \rightarrow 100100 & 101 \rightarrow 101010 \\ 110 \rightarrow 110001 & 111 \rightarrow 111000 \\ \text { d) } E: \mathbf{Z}_{2}^{3} \rightarrow \mathbf{Z}_{2}^{8} & \\ 000 \rightarrow 00011111 & 001 \rightarrow 00111010 \\ 010 \rightarrow 01010101 & 011 \rightarrow 01110000 \\ 100 \rightarrow 10001101 & 101 \rightarrow 10101000 \\ 110 \rightarrow 11000100 & 111 \rightarrow 11100011 \end{array}

In the group \(S_{5}\), let $$ \alpha=\left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 2 & 3 & 1 & 4 & 5 \end{array}\right) \text { and } \quad \beta=\left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 5 & 3 & 4 \end{array}\right) $$ Determine \(\alpha \beta, \beta \alpha, \alpha^{3}, \beta^{4}, \alpha^{-1}, \beta^{-1},(\alpha \beta)^{-1},(\beta \alpha)^{-1}\), and \(\beta^{-1} \alpha^{-1}\)

Why is the set \(\mathbf{Z}\) not a group under subtraction?

a) How many distinct ways are there to paint the edges of a square with three different colors? b) Answer part (a) for the edges of a regular pent脿gon.

Let \(G=\\{q \in \mathbf{Q} \mid q \neq-1\\}\). Define the binary operation \(\circ\) on G by \(x \circ y=x+y+x y\). Prove that \((G, \circ)\) is an abelian group
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.