*in which we meet the quaternions, start exploring matrix groups and act with matrices on vectors and on matrices.
* 798 more words

## Tags » Normal Subgroup

#### Groups Lecture 18

#### Groups Lecture 17

*in which we explore conjugacy classes in and , and prove that is simple.
* 518 more words

#### Groups Lecture 11

*in which we explore groups of order 6 and make a start on quotients.
* 834 more words

#### Groups Lecture 10

*in which we apply group theory to prove Fermat-Euler, use Lagrange to help us find subgroups or determine what a small group must look like, and meet normal subgroups.* 868 more words

#### Proof that any subgroup of index 2 is normal

Let be a subgroup of index 2.

Let and .

If , then , and , hence left coset equals to right coset.

If , then (set minus), and also , thus left coset also equals to right coset. 31 more words

#### Groups Lecture 18

*in which we meet the quaternions and start exploring matrix groups.
* 621 more words

#### Groups Lecture 17

*in which we explore conjugacy classes in and , and prove that is simple.
* 501 more words