Is the intersection of subgroups of G always a subgroup?

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  • Thread starter Chris L T521
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In summary, a subgroup of a group G is a subset of G that satisfies the three conditions of a group: closure, associativity, and existence of an identity element, as well as having inverses for all of its elements. The intersection of two or more subgroups will always be a subgroup of G, as it contains the identity element, is closed under the group operation, and has inverses for all of its elements. However, the intersection can also be the empty set if the subgroups have no common elements, which can still be considered a subgroup of G. The intersection of subgroups and the product of subgroups are different concepts, as the former involves finding common elements while the latter involves multiplying elements and may not always result
  • #1
Chris L T521
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Again, sorry for posting this late. Thanks to those who participated in last week's POTW! Here's this week's problem!

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Problem: Let $H$ and $K$ be subgroups of $G$. Show that $H\cap K$ is a subgroup of $G$. Furthermore, show that this is true for any arbitrary intersection of subgroups of $G$; i.e. if $\{H_{\alpha}\}$ is a collection of subgroups of $G$, then $\bigcap_{\alpha} H_{\alpha}$ is also a subgroup of $G$.

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This week's question was correctly answered by Sudharaka. You can find his solution below.

Theorem: Let \(G\) be a group. The non-empty subset \(H\) of \(G\) is a subgroup of \(G\) if and only if \(ab^{-1}\in H\,\forall\,a,b\in H\).

(Reference: Elementary group theory - Wikipedia, the free encyclopedia)Let \(H\) and \(K\) be subgroups of \(G\). Take any two elements, \(a,b\in H\cap K\). Then, \[a,b\in H\mbox{ and }a,b\in K\]Since \(H\) and \(K\) are subgroups of \(G\), \(ab^{-1}\in H\mbox{ and }ab^{-1}\in K\). Therefore,\[ab^{-1}\in H\cap K\,\forall\,a,b\in H\cap K\]\[\Rightarrow H\cap K\leq G\]Q.E.D.Take any two elements, \(a,b\in \bigcap_{\alpha} H_{\alpha}\mbox{ where }\{H_{\alpha}\}\) is an arbitrary collection of subgroups of \(G\). Then,\[ab^{-1}\in H_{\alpha}\mbox{ for each }H_{\alpha}\in\{H_{\alpha}\}\]\[\therefore ab^{-1}\in\bigcap_{\alpha} H_{\alpha}\,\forall\,a,b\in \bigcap_{\alpha} H_{\alpha}\]\[\Rightarrow\bigcap_{\alpha} H_{\alpha}\leq G\]Q.E.D.
 

FAQ: Is the intersection of subgroups of G always a subgroup?

What is the definition of a subgroup?

A subgroup of a group G is a subset of G that satisfies the three conditions of a group: closure, associativity, and existence of an identity element. Additionally, a subgroup must also satisfy the condition of having inverses for all of its elements.

Is the intersection of two subgroups always a subgroup?

Yes, the intersection of two subgroups H and K of a group G will always be a subgroup of G. This is because the intersection will contain the identity element (as both H and K contain the identity element), will be closed under the group operation (as H and K are both closed under the group operation), and will have inverses for all of its elements (as H and K both have inverses for all of their respective elements).

Is the intersection of multiple subgroups always a subgroup?

Yes, the intersection of any number of subgroups of a group G will always be a subgroup of G. This is because the intersection will contain the identity element, will be closed under the group operation, and will have inverses for all of its elements, as explained in the previous answer.

Can the intersection of subgroups be the empty set?

Yes, it is possible for the intersection of subgroups to be the empty set. This would occur if the subgroups have no elements in common, making their intersection an empty set. However, in this case, the empty set can still be considered a subgroup of G, as it satisfies the three conditions of a group: closure, associativity, and existence of an identity element.

Is the intersection of subgroups the same as the product of subgroups?

No, the intersection of subgroups and the product of subgroups are two different concepts. The intersection of subgroups involves finding the common elements between two or more subgroups, while the product of subgroups involves multiplying the elements of two or more subgroups. The intersection of subgroups will always result in a subgroup, while the product of subgroups may not necessarily result in a subgroup.

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