Purchase Solution

Groups

Not what you're looking for?

Ask Custom Question

Give an example of groups H_i, K_j such that H_1xH_2 is isomorphic to K_1xK_2 and no H_i is isomorphic to any K_j.

Let G be the additive group Q of rational numbers. Show that G is not the internal direct product of any two of its proper subgroups.

If G is the internal direct product of subgroups G_1 and G_2 show that G/G_1 is isomorphic to G_2 and G/G_2 is isomorphic to G_1.

I need detailed rigorous proof please. I have to study for a test.

Purchase this Solution
Solution Summary

This provides examples of working with groups: isomorphic, direct products, and subgroups.

Solution Preview

Problem #1
We consider , , , , then we have
, but is not isomorphic to any .
Problem #2
Proof:
Suppose is the internal direct product of two subgroups and . Then we have and . Now we consider and it can be expressed as . Since both ...

Purchase this Solution
Free BrainMass Quizzes
Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

View More Free Quizzes
Related BrainMass Solutions

  • Electron Donating Groups vs. Electron Withdrawing Groups

    119096 Electron Donating Groups vs. Electron Withdrawing Groups Are there differences between having electron-donating groups and electron-withdrawing groups on benzoate esters? Why or why not? Yes, there is a difference between the two.

  • Perform a Two Population Hypothesis Test

    of groups - 1 c) Within-Groups df = Total df - Between-Groups df viii) Mean Squares (MS) a) Between-Group MS = (Between-Groups SS)/(Between-Groups df) b) Within-Groups MS = (Within-Groups SS)/(Within-Groups df) ix) The F ratio:

  • nonequivalent groups

    36189 Nonequivalent groups What is a major problem with nonequilvalent groups _______ (a) the groups may differ on the dependent measure(s) at the start of the study (b) there may be confounding variables not controlled by random assignment (

  • Theories of Interest Group Behavior

    However, Robert Dahl's pluralist theory stated that their behavior of interest groups could be categorized into three different groups.

  • Characteristics of Interest Groups

    487477 Characteristics of Interest Groups Identify and describe some characteristics of interest groups. Evaluate the reasons why some interest groups are stronger than others. Almost all interest groups share several characteristics.

  • Support Groups: Coping Skills

    Examples of support groups and how they came together to help others and themselves include those groups formed by and managed by hospitals.

  • Stereotypes and how they change overtime

    However, stereotypes can also lead to prejudice and discrimination of certain groups and members of these groups.

  • Key differences between economic and citizens' groups

    The primary difference between economic and citizens' groups is the objectives and intent of these groups. Both are part of civil society, but aim to achieve different means.

  • Write structures that all have the molecular formula C4H8O2

    Ethyl Acetate Functional Groups Ether Ketone Ester (2). 1,4-Dioxane Functional Groups Ether Epoxide (3). Butanoic Acid Functional Groups Carboxylic Acid (4).

  • Functions of status and role play in social interactions?

    367092 What functions do status, role, primary groups, and secondary groups play in social interactions between different racial and ethnic groups. List one example of each.

View More Related Solutions