Group Homomorphism: True or False?

  • MHB
  • Thread starter lemonthree
  • Start date
  • Tags
    Group
In summary, a group homomorphism is a function between two groups that preserves the group operation. To determine if a function is a group homomorphism, you need to check if it preserves the group operation. It can be both one-to-one and onto, but not always. Not every function between two groups is a group homomorphism, as it must preserve the group operation. There are different types of group homomorphisms, such as isomorphisms, endomorphisms, and automorphisms.
  • #1
lemonthree
51
0
Consider the group
(\mathbb{R}^*,\times)
.


"The map
\psi:\mathbb{R}^*\to\mathbb{R}^*
defined by
\psi(x)=|x|
for all
x\in \mathbb{R}^*
is a group homomorphism."

Is this true or false? I'm guessing it's true because
φ (j) = | j |, which means
φ (j * k) = | j * k |
=| j | * | k |
= φ ( j ) * φ ( k ).
 
Physics news on Phys.org
  • #2
Hi lemonthree,

That is true, but you should also check that
  • $\phi(x^{-1}) = \phi(x)^{-1}$
  • $\phi(1) = 1$
 

FAQ: Group Homomorphism: True or False?

1. What is a group homomorphism?

A group homomorphism is a function between two groups that preserves the group structure. This means that the operation in the first group is preserved in the second group. In other words, if we apply the function to two elements in the first group and then perform the group operation, it will be the same as applying the function to the two elements separately and then performing the group operation.

2. How do you determine if a function is a group homomorphism?

To determine if a function is a group homomorphism, we need to check if it preserves the group structure. This means that for any two elements a and b in the first group, the function must satisfy f(a * b) = f(a) * f(b), where * represents the group operation. If this condition is met, then the function is a group homomorphism.

3. Can a function be a group homomorphism if it is not injective or surjective?

No, a function must be both injective (one-to-one) and surjective (onto) to be a group homomorphism. This is because a group homomorphism must preserve the group structure, and if the function is not injective, then different elements in the first group will map to the same element in the second group, which breaks the group structure. Similarly, if the function is not surjective, then there will be elements in the second group that are not mapped to by any element in the first group, also breaking the group structure.

4. Are there different types of group homomorphisms?

Yes, there are different types of group homomorphisms such as monomorphisms, epimorphisms, and isomorphisms. A monomorphism is an injective group homomorphism, an epimorphism is a surjective group homomorphism, and an isomorphism is a bijective group homomorphism. These different types have different properties and are useful for different purposes in group theory.

5. Is the statement "all group homomorphisms are isomorphisms" true or false?

False. While all isomorphisms are group homomorphisms, not all group homomorphisms are isomorphisms. An isomorphism is a bijective group homomorphism, meaning it is both injective and surjective. However, a group homomorphism can be either injective or surjective, but not necessarily both. Therefore, not all group homomorphisms are isomorphisms.

Similar threads

MHB Prove function is a homomorphism
Replies
2
Views
4K
MHB Number of Homomorphisms from $\mathbb{Z}_4$ to $S_4$: A Brief Exploration
Replies
11
Views
2K
I Q about constricting simple lie algebras from simple roots
Replies
1
Views
1K
MHB The automorphic group is abelian
Replies
14
Views
3K
MHB The Symmetry Group of a Hexagonal Prism and Orthogonal Matrices
Replies
1
Views
2K
Valuations and places - decomposition and inertia group
Replies
4
Views
1K
A Classification of reductive groups via root datum
Replies
3
Views
2K
I The way matrices are written without boxes
Replies
9
Views
1K
A Structure of modulo-integer matrix group GL(r,Z(n))?
Replies
17
Views
5K
MHB Symmetry Groups of Cube & Tetrahedron: Orthogonal Matrices & Permutations
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top