Is Every Group with Squared Elements Equal to the Identity Element Abelian?

In summary, the conversation discusses how to prove that a group where the square of every element equals the identity is Abelian. The solution involves showing that a-1 = a and (ab)-1 = ba, which leads to the conclusion that the group is Abelian. The conversation ends with the reminder that feeling stupid is often a part of the learning process.
  • #1
e(ho0n3
1,357
0
[SOLVED] An Abelian Group Problem

Homework Statement
Prove: If G is a group where the square of every element equals the identity element, then G is Abelian.

The attempt at a solution
I've been able to prove is that a-1 = a and that (ab)-1 = ba where a and b are in G. Everything else I've done leads into a dead-end. Any tips?
 
Physics news on Phys.org
  • #2
Okay, you know that a-1= a for every element of the group (including ab) and you know that (ab)-1= ba. What do those two together tell you?
 
  • #3
I see now! Boy do I feel stupid. Thanks.
 
  • #4
Hey, feeling stupid is the beginning of learning!
 

1. What is an Abelian group?

An Abelian group is a mathematical structure that consists of a set of elements and an operation that combines any two elements to form a third element, satisfying certain properties. These properties include closure, associativity, identity element, and inverse element. In an Abelian group, the operation is commutative, meaning that the order in which the elements are combined does not affect the result.

2. What are some examples of Abelian groups?

Some common examples of Abelian groups include the set of integers under addition, the set of real numbers under addition, and the set of complex numbers under addition. Other examples include the set of rational numbers under addition, the set of non-zero integers under multiplication, and the set of positive real numbers under multiplication.

3. What is the difference between an Abelian group and a non-Abelian group?

The main difference between an Abelian group and a non-Abelian group is that in an Abelian group, the operation is commutative, while in a non-Abelian group, the operation is non-commutative. This means that the order in which the elements are combined affects the result in a non-Abelian group, while it does not in an Abelian group.

4. How are Abelian groups used in mathematics?

Abelian groups are used in various areas of mathematics, including abstract algebra, number theory, and geometry. They provide a fundamental structure for understanding symmetry and transformations, and are crucial in the study of symmetry groups and group theory. They also have applications in cryptography, coding theory, and physics.

5. What are some interesting properties of Abelian groups?

One interesting property of Abelian groups is that they form a category, meaning that they can be composed and have homomorphisms between them. Another property is that every subgroup of an Abelian group is also an Abelian group. Additionally, the classification of finitely generated Abelian groups is known as the fundamental theorem of finitely generated Abelian groups, which is a key result in group theory.

Similar threads

Group Theory: Finite Abelian Groups - An element of order
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Proving Normality of [0] in Z/3Z Quotient Group
    • Calculus and Beyond Homework Help
Replies
4
Views
3K
Proving that Aut(K), where K is cyclic, is abelian
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
Fixed point free automorphism of order 2
    • Calculus and Beyond Homework Help
Replies
2
Views
676
Proving Subgroups in Abelian Groups
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
Abelian group as a direct product of cyclic groups
    • Calculus and Beyond Homework Help
Replies
9
Views
2K
Subgroup of Index ##n## for every ##n \in \Bbb{N}##.
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Commutator group in the center of a group
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
Proving that an Abelian group of order pq is isomorphic to Z_pq
    • Calculus and Beyond Homework Help
Replies
5
Views
2K
Showing a property of Abelian groups of order n
    • Calculus and Beyond Homework Help
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top