Lagrange’s Theorem: Cosets, Element Orders, and Subgroup Limits

All flashcards

What is the order of an element $a$ in a group?

The smallest positive integer $k$ such that $a^k=e$, if such an integer exists.

How is the order of an element related to its generated subgroup?

The order of $a$ equals the size of the cyclic subgroup $\langle a\rangle$.

What can be said about a subgroup of index 2?

It is normal in the group.

What is a CLT group?

A finite group with a subgroup of every order dividing its group order.

What does it mean for a group to be simple?

It has no nontrivial proper normal subgroups.

Quiz questions

1. For a finite group $G$ and subgroup $H$, which equation relates the group order, subgroup order, and index?Answer: $|G|=[G:H]|H|$
2. If a finite group $G$ has order 18, what must be true of every element $x\in G$?Answer: $\operatorname{ord}(x)$ divides 18.
3. What follows from Lagrange’s theorem if a group has prime order?Answer: Every such group is cyclic.
4. The group $A_4$ has order 12. What does it show about the converse of Lagrange’s theorem?Answer: A divisor of a group order need not be the order of a subgroup; $A_4$ has no subgroup of order 6.
5. If a prime $p$ divides the order of a finite group, what does Cauchy’s theorem guarantee?Answer: An element of order $p$.