The group operation on G*G is multiplication.

  • Thread starter Justabeginner
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In summary: And once you do know that, the obvious candidate for an isomorphism from G to T should be clear.In summary, we are given the group G and the group A = G * G, where G * G represents the Cartesian product of G with itself. We are asked to prove that the subgroup T = {(g, g)|g ε G} is isomorphic to G. The first step is to define the group operation of A and then find a bijection between G and T that preserves the group operation.
  • #1
Justabeginner
309
1

Homework Statement


Let G be a group, A = G * G. In A, Let T = {(g, g)|g ε G}. Prove that T is isomorphic to G.


Homework Equations





The Attempt at a Solution


A is abelian. Therefore, G * G is abelian. T is a subgroup of G.

I am not sure if my above inferences are even correct. Can someone guide me as to the thought process on this please? Thank you.
 
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  • #2
I don't see how you could have inferred any of those, or why you would need to.

You are asked to show that ##G## and ##T## are isomorphic. There is an obvious candidate for an isomorphism, so you should just verify that it actually is one.
 
  • #3
And for that matter, you haven't told us what G*G means.
 
  • #4
LCKurtz said:
And for that matter, you haven't told us what G*G means.

A = G * G, as in G cross G.
I do not understand how T is directly isomorphic to G.
 
  • #5
Justabeginner said:

Homework Statement


Let G be a group, A = G * G. In A, Let T = {(g, g)|g ε G}. Prove that T is isomorphic to G.


Homework Equations





The Attempt at a Solution


A is abelian. Therefore, G * G is abelian. T is a subgroup of G.
How'd you get A is abelian?

If A is abelian, then obviously GxG is abelian since A=GxG.

How can T be a subgroup of G when it's not a subset of G? It's a subset of A, right?

How did you define the group multiplication for A?

I am not sure if my above inferences are even correct. Can someone guide me as to the thought process on this please? Thank you.
 
  • #6
Justabeginner said:

Homework Statement


Let G be a group, A = G * G. In A, Let T = {(g, g)|g ε G}. Prove that T is isomorphic to G.


Homework Equations





The Attempt at a Solution


A is abelian.

There is nothing in the question to suggest this. If [itex]G[/itex] is not abelian then [itex]A[/itex] will not be abelian.

Therefore, G * G is abelian. T is a subgroup of G.

Actually T is a subgroup of A.

I am not sure if my above inferences are even correct. Can someone guide me as to the thought process on this please?

You need to find a bijection [itex]\phi : G \to T[/itex] such that [itex]\phi(g)\phi(h) = \phi(gh)[/itex] for every [itex]g \in G[/itex] and [itex]h \in G[/itex].

It would be good to start by writing out the group operation of A, and see what happens when you restrict it to T.
 
  • #7
LCKurtz said:
And for that matter, you haven't told us what G*G means.

Justabeginner said:
A = G * G, as in G cross G.
I do not understand how T is directly isomorphic to G.

What is the group operation on G*G? You have to know that before you can even talk about an isomorphism.
 

1. What is an isomorphism in mathematics?

An isomorphism in mathematics is a type of function between two mathematical structures that preserves their structure. This means that the function maps elements of one structure to elements of the other structure in a way that maintains their properties and relationships.

2. What does it mean for an isomorphism to be abelian?

An isomorphism is considered abelian if it preserves the commutative properties of the structures it is mapping between. This means that the order in which operations are performed in one structure is the same as the order in which they are performed in the other structure.

3. Why is the concept of isomorphisms important in mathematics?

Isomorphisms allow mathematicians to study and understand different mathematical structures by relating them to each other. This can reveal underlying patterns and connections between seemingly unrelated structures, leading to new insights and discoveries in mathematics.

4. How can I determine if two structures are isomorphic?

To determine if two structures are isomorphic, you can look for a function that maps elements of one structure to elements of the other structure in a way that preserves their properties and relationships. You can also check if the structures have the same number of elements and if their operations are performed in the same order.

5. Can an isomorphism exist between non-abelian structures?

Yes, an isomorphism can exist between non-abelian structures. The key requirement for an isomorphism is that the function preserves the structure of the two structures being mapped, regardless of whether they are abelian or non-abelian.

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