MHB Proving a Subset of a Group is a Group

  • Thread starter Thread starter cbarker1
  • Start date Start date
  • Tags Tags
    Group
cbarker1
Gold Member
MHB
Messages
345
Reaction score
23
Dear Everyone,

I want to show that a subset of a group is still a group by using the subgroup criterion which states that a subset $H$ of a group $G$ is a subgroup if and only if $H \ne \emptyset$ and for all $x,y \in H, xy^{-1}\in H$. I am having trouble how to show that criterion in the following exercise:

"Let $(G,\star)$ be a group and $H$ be a nonempty subset of $G$ such that $H$ is closed under the group operation $\star$ and is closed under inverses. That is, for all $h_{1}$ and $h_{2}$ in $H$, both $h_{1} \star h_{2} \in H$ and $h^{-1} \in H$. Prove that $H$ is a group under the operation $\star$ restricted to $H$."

My attempt:

We know that the $H$ is not the empty set. We know that $h^{-1}\in H$ because $H \subset G$. So let $h_{1}, h_{2} \in H$ be arbitrary. Let $h_{2}=h^{-1}$. Then $h_{1}\star h^{-1} \in H$.

Thanks,
Cbarker1
 
Physics news on Phys.org
Hi Cbarker1,

You have effectively shown the forward direction.
That is, if $H$ is a subgroup, then for all $x,y \in H: xy^{-1}\in H$.
Indeed that follows because $y^{-1}$ is in $H$ as it is closed under inverses.
And $x\star y^{-1}$ is subsequently in $H$ because $H$ is closed under the group operation.

However, to prove the criterion, we also need to prove it in the reverse direction.
So if $H\ne\varnothing$ and for all $x,y \in H: x \star y^{-1}\in H$, does it follow that $H$ is closed under the group operation? And also closed under inverses?
 
Hi Cbarker1.

Cbarker1 said:
We know that the $H$ is not the empty set. We know that $h^{-1}\in H$ because $H \subset G$. So let $h_{1}, h_{2} \in H$ be arbitrary. Let $h_{2}=h^{-1}$. Then $h_{1}\star h^{-1} \in H$.

Thanks,
Cbarker1
You’re not quite there yet; all you’ve shown is that $h_1\star h_2\in H$ whereas you want to show that $h_1\star h_2^{-1}\in H$. You need one more step: that $h_2\in H$ implies $h_2^{-1}\in H$ (as $H$ is closed under inverses). Now you can apply closure under $\star$ to $h_1$ and $h_2^{-1}$.

To prove the criterion from the reverse assumption, first show that the identity $e$ is in $H$. (Hint: take $h\in H$ (you can do this as $H$ is nonempty) and apply the criterion to $h,h$.) Then, to show that the inverse of any $h\in H$ is in $H$, apply the criterion to $e,h$, and to show that $h_1\star h_2\in H$ for any $h_1,h_2\in H$, apply the criterion to $h_1,h_2^{-1}$.
 
Last edited:
I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...

Similar threads

Replies
13
Views
561
Replies
9
Views
1K
Replies
3
Views
428
Replies
5
Views
2K
Replies
1
Views
2K
Replies
19
Views
2K
Replies
2
Views
1K
Back
Top