Yes, it should be |G'|. Thank you for catching that!

In summary, group theory discusses the concept of group homomorphisms and their properties. It is shown that if the group G is finite, then the cardinality of the image of G under a homomorphism, phi(G), is also finite and is a divisor of the cardinality of G. This is proven using Lagrange's Theorem and the first isomorphism theorem.
  • #1
ehrenfest
2,020
1
[SOLVED] group theory

Homework Statement


Let [itex]\phi:G \to G'[/itex] be a group homomorphism. Show that if |G| is finite, then [itex]|\phi(G)|[/itex] is finite and is a divisor of |G|.

Homework Equations


The Attempt at a Solution


Should the last word be |G'|? Then it would follow from Lagrange's Theorem.
 
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  • #2
Nope; it's right as stated. (And can also use Lagrange's theorem in its proof)
 
  • #3
Think first isomorphism theorem.
 
  • #4
I haven't gotten to the first isomorphism theorem yet, but I don't even need it:

We know that [itex]\phi^{-1}(\phi(a)) = aH = Ha[/itex], where H = Ker(phi). So, the cardinality of phi(G) will be the index of H in G, which must divide |G| by Lagrange's Theorem.

Is that right?
 
  • #5
Bingo.
 

1. What is group theory?

Group theory is a branch of mathematics that studies the algebraic structures known as groups. A group is a set of elements that can be combined together using a binary operation (such as multiplication or addition) that follows specific rules.

2. What are the applications of group theory?

Group theory has applications in various fields such as physics, chemistry, computer science, and cryptography. It can be used to study symmetry in physical systems, understand chemical reactions, and analyze algorithms in computer science.

3. What are the basic properties of a group?

A group must satisfy four basic properties: closure, associativity, identity, and inverse. Closure means that when two elements of the group are combined using the binary operation, the result is also an element of the group. Associativity means that the grouping of elements in the operation does not change the result. Identity means that there exists an element in the group that when combined with any other element results in that element. Inverse means that for every element in the group, there exists another element that when combined results in the identity element.

4. How is group theory related to symmetry?

Group theory is closely related to symmetry because it provides a mathematical framework for understanding and analyzing symmetrical objects and systems. In group theory, symmetries are represented as elements in a group, and the properties of the group can help determine the possible symmetries of an object or system.

5. What are some common examples of groups?

Some common examples of groups include the group of integers under addition, the group of real numbers (excluding 0) under multiplication, and the group of symmetries of a square. Other examples can be found in matrix operations, permutations, and modular arithmetic.

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