Please Login to access more options.

Problem 55 (The Union Of Two Subgroups)

Suppose that $G$ is a group and that $H$ and $K$ are subgroups of $G$. Is $H\cup K$ a subgroup of $G$? Either prove that is, or find a counterexample.

Solution

Let's show that the union of two subgroups of a group is not always a subgroup. Let $G=\mathbb{Z}_6$ and $H=\{0,3\} $ and $K=\{0,2,4\}$. Note that both $H$ and $K$ are subgroups of $G$ because they are both nonempty subsets of $G$ that are closed under the operation and under taking inverses. However, their union $A=H\cup K =\{0,2,3,4\}$ is not closed under the operation as $1=3+4$ is not in $A$. Thus the union of subgroups is not necessarily a subgroup.

Tags

Change these as needed.

  • Week7 - Which week are you writing this problem for?
  • Ben - Sign your name by just changing "YourName" to your wiki name.
  • Complete
  • When you are ready to submit this written work for grading, add the phrase [[!Submit]] to your page. This will tell me that you have completed the page (it's past rough draft form, and you believe it is in final draft form). Don't type [[!Submit]] on a rough draft.
  • If I put [[!NeedsWork]] on your page, then your job is to review what I've written, address any comments made, and then delete all the comments I made. When you have finished reviewing your work, leave [[!NeedsWork]] on your page and type [[!Submit]]. (Both tags will show up). This tells me you have addressed the comments.
  • I'll mark your work with [[!Complete]] after you have made appropriate revisions.