Finite Group Proof: Proving H is a Subgroup of G

In summary, To prove that a subset H of a finite group G is a subgroup, it must be shown that it is closed, contains the identity element of G, and contains the inverse of each of its elements. The existence of inverse is a sufficient condition for H to be a subgroup, as it guarantees closure and the presence of the identity element. Additionally, since H is a finite group, there must exist a power n such that a^n=e for any element a in H, making the inverse of a simply a^{n-1}.
  • #1
kathrynag
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Homework Statement



Let G be a finite group andd H a subset of G. Prove H is a subgroup of G iff H is closed.

Homework Equations





The Attempt at a Solution


Let G be a finite group and H be a subgroup.
G is a finite group, therefore it is closed, has an inverse and has an identity.
We want to show H is only a subgroup of G iff H is closed.
To be a subgroup, H must be closed, contain the identity element of G, and contain the inverse.

Now I'm stuck.
 
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  • #2
For H to be a subgroup of G existence of inverse is sufficient condition.Because if [tex]a \epsilon\ G[/tex] then [tex] a^{-1}
\epsilon G[/tex] due to existence of inverse.But since H is closed [tex]a*a^{-1} \epilson \ H[/tex].Therefore [tex]e \epilson H[/tex] .
Now, since H is a finite group there must exist an n such that [tex]a^n=e[/tex], otherwise it will be an infinite group.So for any [tex]a\epsilon H[/tex] , the inverse is [tex]a^{n-1}[/tex].
 
Last edited:
  • #3
Wow, that makes more sense now.
 

1. What is a finite group?

A finite group is a mathematical structure that consists of a finite set of elements and a binary operation that combines any two elements to form a third element in the set. It is a fundamental concept in abstract algebra and has many applications in various fields of science and mathematics.

2. What is a subgroup?

A subgroup is a subset of a group that itself forms a group under the same binary operation. In other words, a subgroup is a smaller group that retains the same algebraic structure as the larger group.

3. What is a finite group proof?

A finite group proof is a mathematical argument that demonstrates the existence of a finite group by showing that a given set of elements and binary operation fulfills all the necessary properties of a group, such as closure, associativity, identity, and inverse elements.

4. How do you prove that H is a subgroup of G?

To prove that H is a subgroup of G, you need to show that H is a subset of G, and that the binary operation on H is the same as the binary operation on G. Additionally, you must demonstrate that H satisfies the four group axioms: closure, associativity, identity, and inverse elements.

5. What are some common techniques used in finite group proofs?

Some common techniques used in finite group proofs include direct proof, where each property of a group is verified individually, and subgroup test, where the subgroup is shown to satisfy the necessary conditions for being a subgroup. Other methods include proof by contradiction and proof by induction.

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