Abstract algebra class equation

Click For Summary

Homework Help Overview

The discussion revolves around a problem in abstract algebra concerning the properties of group orders, specifically focusing on groups where each element has an order that is a power of a prime number. The participants are exploring the implications of this property on the overall order of the group.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants express uncertainty about how to begin tackling the problem and mention the use of the class equation. Questions arise regarding the definitions of "order of a group" and "order of an element," with suggestions to consider specific examples for clarity.

Discussion Status

The discussion is ongoing, with participants seeking clarification on definitions and exploring foundational concepts. Some guidance has been offered, such as applying Cauchy's theorem, but no consensus has been reached regarding the approach to the problem.

Contextual Notes

There is a noted emphasis on definitions from the textbook, and the participants are considering the implications of the order of the group not being a power of p, which may influence their understanding of the problem.

Mr Davis 97
Messages
1,461
Reaction score
44
1. The problem statement, all variables and given/known
If each element of a group, G, has order
which is a power of p, then the order of G is also a prime power.

Homework Equations

The Attempt at a Solution


I am not sure really where to get started. I know that the class equation will be used though
 
Physics news on Phys.org
Mr Davis 97 said:
1. The problem statement, all variables and given/known
If each element of a group, G, has order
which is a power of p, then the order of G is also a prime power.

Homework Equations

The Attempt at a Solution


I am not sure really where to get started. I know that the class equation will be used though
How does your textbook define the phrase "order of a group"?
 
Mark44 said:
How does your textbook define the phrase "order of a group"?
The order of G is the number of elements in G
 
And what is the order of an element of a group? It might be helpful to look at some examples, such as ##(\mathbb{Z_4}, *)## or ##(\mathbb{Z_5}, *)##.
 
The order of an element of a group is the order of the cyclic subgroup that it generates.
 
Mr Davis 97 said:
The order of an element of a group is the order of the cyclic subgroup that it generates.

Sure it is. I suggest you apply Cauchy's theorem to your group. Suppose the order of ##G## is NOT a power of ##p##?
 

Similar threads

Undergrad Meaning of "passage to the quotient"?
  • · Replies 3 ·
Replies
3
Views
1K
A detail in a proof about isomorphism classes of groups of order 21
  • · Replies 5 ·
Replies
5
Views
2K
Abstract Algebra, order of ab is equal to the order of a times the order of b?
  • · Replies 3 ·
Replies
3
Views
10K
Proving something is commutative in abstract algebra
  • · Replies 12 ·
Replies
12
Views
3K
Abstract Algebra - Group of Order 12 with Conjugacy Class of Order 4
  • · Replies 2 ·
Replies
2
Views
5K
Undergrad Localising a non integral domain at a prime
  • · Replies 17 ·
Replies
17
Views
3K
Abstract algebra Polynomials and Prime
  • · Replies 3 ·
Replies
3
Views
2K
Using class equation to show that intersection is nontrivial
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
1K
Abstract Algebra: Bijection, Isomorphism, Symmetric Sets
  • · Replies 2 ·
Replies
2
Views
2K