If G, H, and K are groups and G = H K, prove that G contains

Question:

If G, H, and K are groups and G = H × K, prove that G contains subgroups that are isomorphic to H and K.

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

Step by Step Answer:

Related Book For book-img-for-question

Discrete and Combinatorial Mathematics An Applied Introduction

ISBN: 978-0201726343

5th edition

Authors: Ralph P. Grimaldi

See More Books

Question Posted: Jun 21, 2016 09:11 AM

See More Questions

If G = (Z6, +), H = (Z3, +), and K = (Z2, +), find an isomorphism for the groups H K and G.
(a) Prove that the function h in Example 17.10 is one-to- one and onto and preserves the operation of addition. (b) Let (F, +, ) and (K, , ) be two fields. If g: F K is a ring isomorphism and a is a...
If G = (V, E) is an undirected graph with |V|-n and |E| = k, the following matrices are used to represent G. Let V = {v1, v2, . . . , Define the adjacency matrix A = (aI j)nÃn where aIJ = 1 if...
Show by approximating with integrals that the number of distinct triples of integers between 0 and $n$ is about $n^{3} / 6$.
What should Willie do? It was late October of 2009, and Willis Willie Stahe was running late. As president of Midwest Universitys Computing and Information Systems student club, he had helped...
61. The solubility product of CuS, AgS and HgS are 10 37, 104 and 1054 respectively. The solubility of these sulphides will be in the order (1) HgS> AgS > Cus (2) AgS>HgS > CuS (3) CuS AgS > HgS (4)...
What roles might more recent analysis activities, such as data mining, predictive statistical analyses, and operational experiments, play in increasing organizational effectiveness?
Classic Cabinets has one factory in which it produces two product lines. Walter manages the Wood Division, which produces wood cabinets, and Mary manages the Metal Division, which produces metal...
1. What is the best description of safety capacity? 2. If throughput is 20 per hour and the capacity is 30 per hour, what is the utilization? 3. What is the importance of safety capacity to a manager...
Consider a system that allocates pages of different sizes to its processes. What are the advantages of such a paging scheme? What modifications to the virtual memory system are provide this...
Let f: G H be a group homomorphism with eH the identity in H. Prove that (a) K = {x G|f(x) = eH] is a subgroup of G. (K is called the kernel of the homomorphism.) (b) if g G and x K, then gxg-1 ...
Let G be a group where a2 = e for all a G. Prove that G is abelian.
A stockroom worker pushes a box with mass 11.2 kg on a horizontal surface with a constant speed of 3.50 m/s. The coefficient of kinetic friction between the box and the surface is 0.20. (a) What...
Super PB is the best-selling peanut butter on the market. A new peanut butter has come to market at half the price of Super PB and is affecting the demand for Super PB. Which determinant of demand is...
What principles from GCU's mission, vision, and Christian heritage that you read about in this week's readings can you implement in your career, personal or spiritual life?
Explore the concept of ecological stoichiometry and its significance in understanding nutrient cycling dynamics, energy flow, and ecosystem productivity across terrestrial and aquatic ecosystems.
What are the ramifications of biotic homogenization on global biodiversity patterns, and how do invasive species exacerbate this phenomenon through niche displacement and competitive exclusion ?
Does the salinity measured by the refractometers confirm or counter the measurements made by the hydrometers?
Define accounting return variance analysis and describe when it is used.
Nitrogen monoxide reacts with hydrogen as follows: 2NO(g)+ H2(g) N2O(g) + H2O(g) The rate law is [H2]/ t = k[NO]2[H2], where k is 1.10 107 L2/(mol2s) at 826oC. A vessel contains NO and H2 at...
The Pistons and the Bulls are tied, and time has run out in the game. However, the Pistons have a player at the free throw line, and he has two shots to make. He generally makes 83% of the free throw...
What is the probability that there are exactly two girls in a family with four children? Assume that girls and boys are equally likely.
The table at right gives numbers of students in several categories. Are the events "10th grade" and "Female" dependent or independent? Explain your reasoning. Male Female Total 10th grade 263 249 512...
The following financial information is given. Year 1 Year 2 Book value of assets $18,000 $26,000 Market value of equity 18,000 60,000 12 months ended Year 1 12 months ended Year 2 Sales $1,000 $1,300...
Purple Corporation is undergoing financial difficulty and planning to liquidate its assets. Current ratio is currently at 1.5x. Purple Corp. current liabilities is 500,000; non-current liabilities of...
Amor Co. is undergoing financial difficulty and the shareholders decided to liquidate the company. The company reported cash of P65,000 and building valued at P875,000. They owe P560,000 to various...

If G, H, and K are groups and G = H K, prove that G contains

Question:

Step by Step Answer:

Let denote the operation in G H and K Let S h 0h H Here 0 is the identity for H an...View the full answer

Discrete and Combinatorial Mathematics An Applied Introduction

Students also viewed these Linear Algebra questions