/*! 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 7 Let \(f: G \rightarrow H\) be a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 7

Let \(f: G \rightarrow H\) be a group homomorphism onto \(H\). If \(G\) is a cyclic group, prove that \(H\) is also cyclic.

Short Answer

Expert verified
Given a group \(G\) that is cyclic and a group homomorphism \(f: G \rightarrow H\), the group \(H\) is also cyclic, as every element of \(H\) can be represented as \(h^n = (f(g))^n = f(g^n)\), a power of a single element, i.e., it is generated by a single element.

Step by step solution

01

Define Generators

Start by defining the generator for the cyclic group \(G\). Let's say \(G\) is generated by a single element \(g\). This means every element in \(G\) can be written in the form \(g^n\), for some integer \(n\).
02

Apply Group Homomorphism

Then apply \(f\) to both sides. We have \(f(g^n) = (f(g))^n\) on the right because \(f\) is a homomorphism and preserves the group operation, so applying \(f\) maintains the exponent.
03

Define H Generators

Now let's say \(h = f(g)\) is an element in \(H\). Each element of \(H\) is \(h^n = (f(g))^n = f(g^n)\), by the properties of a homomorphism and definition of our generator \(h\). This means \(H\) is generated by \(f(g)\) or \(h\), where \(f(g)\) is an element of \(H\).
04

Conclude H is Cyclic

Based on the previous step, we can conclude that \(H\) is a cyclic group, as it can be generated by a single element.

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 \(G=\left(\mathbf{Z}_{24},+\right)\), find the cosets determined by the subgroup \(H=\\{[3]\rangle\). Do likewise for the subgroup \(K=\langle[4]\rangle\).

The \((5 m, m)\) five-times repetition code has encoding function \(E: \mathbf{Z}_{2}^{m} \rightarrow \mathbf{Z}_{2}^{5 m}\), where \(E(w)=\) wwwww. Decoding with \(D: \mathbf{Z}_{2}^{5 m} \rightarrow \mathbf{Z}_{2}^{m}\) is accomplished by the majority rule. (Here we are able to correct single and double errors made in transmission.) a) With \(p=0.05\), what is the probability for the transmission and correct decoding of the signal 0 ? b) Answer part (a) for the message 110 in place of the signal \(0 .\) c) For \(m=2\), decode the received word \(r=0111001001\). d) If \(m=2\), find three received words \(r\) where \(D(r)=00\). e) For \(m=2\) and \(D: \mathbf{Z}_{2}^{10} \rightarrow \mathbf{Z}_{2}^{2}\), what is \(\left|D^{-1}(w)\right|\) for each \(w \in \mathbf{Z}_{2}^{2}\) ?

Verify that \(\left(Z_{p}^{*}, \cdot\right)\) is cyclic for the primes 5,7, and 11 .

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

For each of the following sets, determine whether or not the set is a group under the stated binary operation. If so, determine its identity and the inverse of each of its elements. If it is not a group, state the condition(s) of the definition that it violates. a) \(\\{-1,1\\}\) under multiplication b) \(\\{-1,1\\}\) under addition c) \(\\{-1,0,1\\}\) under addition d) \(\\{10 n \mid n \in \mathbf{Z}\\}\) under addition e) The set of all functions \(f: A \rightarrow A\), where \(A=\\{1,2,3,4\\}\), under function composition f) The set of all one-to-one functions \(g: A \rightarrow A\), where \(A=\\{1,2,3,4\\}\), under function composition g) \(\left\\{a / 2^{n} \mid a, n \in \mathbf{Z}, n \geq 0\right\\}\) under addition
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure 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.