How can inverses be proven to exist in a group under *?

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In summary: Pick any element g and define f':G \to G . Show that f is an injective map. Since G is finite that makes it a bijection. So there exists an element e such that ge=g. Can you show eg=g? Can you show eh=he=h for ANY element of G, not just g?Get inverses the same way you got an identity. Pick any element g and define f':G \to G . Show that f is an injective map. Since G is finite that makes it a bijection. So there exists an element e such that ge=g. Can you show eg=g? Can you show eh=he=h for ANY element of G,
  • #1
AdrianZ
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Homework Statement



Let G be a finite nonempty set with an operation * such that:
1. G is closed under *.
2. * is associative.
3. Given a,b,c in G with a*b=a*c, then b=c.
4. Given a,b,c in G with b*a=c*a, then b=c.

Prove that G must be a group under *.

The Attempt at a Solution



It's obvious that identity element satisfies the conditions 3 and 4, but I don't know whether that proves that the identity element is contained in G or not? moreover, How can I show that the inverse of any element in G is contained in G?
 
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  • #2


You have to prove there IS an identity first. Start by picking any element g and defining f(x)=gx. Show using cancellation that f is an injective map from G->G. Since G is finite that makes it a bijection. So there exists an element e such that ge=g. Can you show eg=g? Can you show eh=he=h for ANY element of G, not just g?
 
  • #3


Dick said:
Start by picking any element g and defining f(x)=gx. Show using cancellation that f is an injective map from G->G.
let's take [itex]f(x_1)=f(x_2)[/itex], hence, [itex]g*x_1=g*x_2[/itex] and the third axiom implies [itex]x_1=x_2[/itex]. that proves f is injective.
Since G is finite that makes it a bijection.
if I've understood you correctly then I disagree with this part, f can be injective and finite, and yet it fails to be surjective. It's obvious from the definition of f that it's bijective though, so let's continue.

So there exists an element e such that ge=g.
sounds fine.

Can you show eg=g?
sure, I just need to define [itex]f':G \to G , f(x)=xg[/itex] and prove that f' is bijective. true?

Can you show eh=he=h for ANY element of G, not just g?
well, g was no sacred element and we had stated 'pick any element g in G'. Doesn't that already suffice to conclude that? all we need to do is that we should define a new f. so if we denote the dependence of f on g by [itex]f_g: G \to G[/itex] that would be fine. right?

OK. so now it's also easy to verify that the inverse of any element exists, It deploys the same logic about bijectivity but I'm failing at formalizing it. Could you help please?
 
  • #4


AdrianZ said:
let's take [itex]f(x_1)=f(x_2)[/itex], hence, [itex]g*x_1=g*x_2[/itex] and the third axiom implies [itex]x_1=x_2[/itex]. that proves f is injective.

if I've understood you correctly then I disagree with this part, f can be injective and finite, and yet it fails to be surjective. It's obvious from the definition of f that it's bijective though, so let's continue.


sounds fine.


sure, I just need to define [itex]f':G \to G , f(x)=xg[/itex] and prove that f' is bijective. true?


well, g was no sacred element and we had stated 'pick any element g in G'. Doesn't that already suffice to conclude that? all we need to do is that we should define a new f. so if we denote the dependence of f on g by [itex]f_g: G \to G[/itex] that would be fine. right?

OK. so now it's also easy to verify that the inverse of any element exists, It deploys the same logic about bijectivity but I'm failing at formalizing it. Could you help please?

f is bijective because G is finite. Since f is injective the image f(G) contains the same number of elements as G and is contained in G. And sure, for any two elements of the group g and h you can find elements such that g*e_g=g and h*e_h=h. But now you have to show e_g=e_h.
 
  • #5


Dick said:
f is bijective because G is finite. Since f is injective the image f(G) contains the same number of elements as G and is contained in G.
Alright. I forgot that f was from G to G. that's right.
And sure, for any two elements of the group g and h you can find elements such that g*e_g=g and h*e_h=h. But now you have to show e_g=e_h.
Now I see what you mean. makes sense. I'll think about it.
How about proving the existence of inverses?
 
  • #6


AdrianZ said:
Alright. I forgot that f was from G to G. that's right.

Now I see what you mean. makes sense. I'll think about it.
How about proving the existence of inverses?

Get inverses the same way you got an identity.
 

1. What is a group in mathematics?

A group is a mathematical structure consisting of a set of elements and an operation (usually denoted by *) that combines any two elements to produce a third element. The operation must also satisfy certain properties, including closure, associativity, identity, and inverse.

2. How do you prove that G is a group under *?

To prove that G is a group under *, you must show that the operation * satisfies the four group properties: closure, associativity, identity, and inverse. This can be done by demonstrating that for any elements a, b, and c in G, (a * b) * c = a * (b * c), there exists an identity element e such that a * e = e * a = a, and every element a has an inverse element a^-1 such that a * a^-1 = a^-1 * a = e.

3. What are some common examples of groups?

Some common examples of groups include the integers under addition, the real numbers (excluding 0) under multiplication, the symmetries of a geometric figure, and the set of invertible matrices under matrix multiplication.

4. Can a group have more than one operation?

Yes, a group can have more than one operation as long as each operation satisfies the four group properties. In this case, the group is referred to as a groupoid or a multipliable group.

5. How does the concept of a group apply to other fields of study?

The concept of a group is not limited to mathematics and can be applied to other fields of study, such as physics, chemistry, and computer science. In physics, groups are used to describe symmetries and conservation laws. In chemistry, groups are used to classify and predict the properties of molecules. In computer science, groups are used in cryptography and coding theory.

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