Subgroups: Tests, Generation, Cosets, and Lagrange's Theorem

All flashcards

Left coset

For a subgroup $H$ and group element $a$, the set $aH=\{ah:h\in H\}$.

Index of a subgroup

The number of distinct left cosets of the subgroup in the group.

Lagrange's theorem

For finite $G$ and subgroup $H$, $[G:H]=|G|/|H|$.

Generated subgroup

The smallest subgroup containing a specified subset of a group.

Normal subgroup

A subgroup whose left and right cosets agree for every group element.

Quiz questions

1. Which condition is equivalent to the subgroup test for a subset $H$ of a group $G$?Answer: $H$ is nonempty and $ab^{-1}\in H$ for every $a,b\in H$.
2. A finite nonempty subset of a group is closed under products. What can be concluded?Answer: It is a subgroup; finiteness ensures its elements have inverses in the subset.
3. When is the union of two subgroups of the same group a subgroup?Answer: Exactly when one of the two subgroups is contained in the other.
4. For a finite group $G$ and subgroup $H$, which formula gives the index of $H$?Answer: $[G:H]=|G|/|H|$.
5. What condition defines a normal subgroup $H$ of a group $G$?Answer: $aH=Ha$ for every $a\in G$.