Show that if a finite group G has exactly one subgroup H of a given order, then

Question:

Show that if a finite group G has exactly one subgroup H of a given order, then H is a normal subgroup of G.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

A First Course In Abstract Algebra

ISBN: 9780201763904

7th Edition

Authors: John Fraleigh

See More Books

Question Posted: Jan 02, 2023 04:46 AM

See More Questions

Show that if a finite group G contains a nontrivial subgroup of index 2 in G, then G is not simple.
Show that if a simple graph G has k connected components and these components have n1, n2, . . . , nk vertices, respectively, then the number of edges of G does not exceed C(n,2)
Show that if a G, where G is a finite group with identity e, then there exists n Z + such that a n = e.
Read the case study about Joy Jowie Inc and write a detailed paper about it
How would you rate the privacy protections available to individuals who purchase items through Amazon.com? How safe are similar data of individuals who have never purchased an item through...
Shown below is a recent income statement for B-D Electric. B-D ELECTRIC Income Statement For the Year Ended January 31, Year 2 Net sales $ 7,500,000 Less: Cost of goods sold 4,100,000 Gross profit $...
How is financial accounting different from management accounting?
An alert hiker sees a boulder fall from the top of a distant cliff and notes that it takes 1.30 s for the boulder to fall the last third of the way to the ground. You may ignore air resistance. (a)...
In a single purchase crab length of handle is 600 mm and diameter of load drum is 200 mm. Number of teeth on pinions are 20 and that of spur wheel are 100. Find velocity ratio. On this machine, 100 N...
Select two different perfume advertisements for any brand of perfume. Design and conduct an experiment to determine which ad is more effective. Use a student sample with 10 students being exposed to...
Give an example of a nontrivial homomorphism for the given groups, if an example exists. If no such homomorphism exists, explain why that is so. : Z 12 Z 5
Let G be a group. An element of G that can be expressed in the form aba -1 b -1 for some a, b G is a commutator in G. The preceding exercise shows that there is a smallest normal subgroup C of a...
Public policy is necessary to protect the average citizen from the power of vested interest groups. In the absence of government intervention, regulated industries such as airlines, railroads, and...
1. Basic economics focus on Productivity, Opportunity Cost, Choices, Price Incentives, Specialization and Voluntary Exchange. Briefly explain each of these key terms in basic economics. 2. It is...
Suppose the demand for physician visits is p = 150-3Q. Suppose physicians charge $75 per visit. a. How many times will a patient visit a physician if the patient does not have any insurance? b. Find...
Nano Electronics is forecasted to pay a 5.00 dividend at the end of year one, a 5.50 dividend at the end of year two and a 6.05 dividend at the end of year three. In addition, at the end of the third...
The sprung parts of a vehicle weigh 9.79 KN, its center of gravity is 106.7 cm behind the front axle, and the wheelbase is 228.6 cm. The combined stiffness of the springs of the front suspension is...
Consider a small open economy (e.g. the Netherlands) producing two goods, clothing and food. The clothing industry uses capital (K) and labor (LC) as inputs, while the food industry uses land (La)...
Consider the mapping X defined on ((, A) onto (' = X((), the image of ( under X, and Let C' ( P ((') be defined as follows: C' = {B ( ('; B = X(A), A ( A}. Then, by means of an example, show that C'...
Distinguish among total-moisture content, free-moisture content, equilibrium-moisture content, unbound moisture, and bound moisture.
Answer Problem 14.39 under this definition of success? To reduce variability, the investigators also considered a criterion of at least 50% improvement on two successive visits as a definition of...
Divide participants according to median log 10 CO (adjusted), and estimate survival curves for each subgroup?
A person is selected randomly from the population and followed for 1.5 years. If the true rate of allergic reactions is 5 reactions per 100 person-years, what is the probability that the subject will...
1. You are recently decided to invest $5,000 in the stock market. With such a small amount of money, you decide to invest in two stocks, allowing you easy access to a diversified portfolio. Relevant...
In the process of saving some money for my kids college I plan to make six annual deposits of $4000 into a secret savings account that pays an interest of 4% compounded annually. Two years after...
1. Possibly in a group discussion, build a scenario relevant for the motor vehicle industry in ten years time. How will people be using their cars? What will they expect in terms of size,...

Show that if a finite group G has exactly one subgroup H of a given order, then

Question:

Step by Step Answer:

Let g G Because the inner automorphism i g G G is a onetoo...View the full answer

A First Course In Abstract Algebra

Students also viewed these Mathematics questions