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Definition (Subgroup)
Let $(G,\cdot)$ be a group, and let $H$ be a nonempty subset of $G$. Then $H$ is called a $\textdef{subgroup}$ of $G$ if the following hold:
- $\textbf{[Closure]}$ for all $h,k\in H$ we have $h\cdot k\in H$, and
- $(H,\cdot)$ is a group.
When $H$ is a subgroup of $G$, we write $H\le G$. Any subgroup of $G$ that is not equal to $G$ itself we call a $\textdef{proper subgroup}$. The subset of $G$ consisting of just the identity we call the $\textdef{trivial subgroup}$.
The following pages link to this page.
- Definition.Subgroup
- Definition.SubgroupGeneratedByAnElement
- Problem.SubgroupsAreSubsetsThatAreClosedUnderProductsAndInverse
- Schedule.20131014
- Schedule.20131018
- Schedule.20131021
- Schedule.20161010
- Schedule.20161012
- Schedule.20161014
- Schedule.20161017
- Schedule.20161021
- Schedule.20171013
- Schedule.20171016
- Schedule.20171020
- Schedule.20171023
- Schedule.20171025
- Schedule.20171027
- Schedule.AllProblems
- Schedule.InProgressOld