## Normal Subgroups and Quotient Groups

Welcome to our activities for Week 10 (the week of November 7).Our theme this week is

*normal subgroups*and

*quotient groups*. We have already encountered the concept of a normal subgroup in the second homework sheet. A subgroup of a group

*G*is

*normal*if it is a union of conjugacy classes in

*G*, or equivalently if the left and right cosets of the subgroup determined by every element of the group are the same set. This looks like a fussy technical property, and it might not be immediately obvious why it is of interest. The reason(s) why normal subgroups are considered to be interesting and important relates to a question that might arise from thinking about Lagrange’s Theorem. Lagrange’s Theorem says that if

*G*is a group of order

*n*, and

*H*is a subgroup of

*G*of order

*m*, then the number

*n/m*is an integer (call it

*k*). Given this surprising relationship, one might wonder if we can somehow “divide” the group

*G*by the subgroup

*H*and get a group of order

*k*. The answer to that is no. But, if the subgroup

*H*is

*normal*in

*G*, the answer is sort of yes, although we don’t use the term “divide” (but we do use the related term

*quotient*). In this siuation we can define a group of order

*k*that arises from the situation of

*H*as a normal subgroup of

*G*. Its elements are the (left or right) cosets of

*H*in

*G*, and it is called the

*quotient group G/H*(

*G*modulo

*H*). We have already kind of seen it in our discussion in Week 10 of how the image of a group homomorphism relates to the domain and the kernel.

The reason(s) why normal subgroups are considered to be interesting and important relates to a question that might arise from thinking about Lagrange’s Theorem. Lagrange’s Theorem says that if

*G*is a group of order

*n*, and

*H*is a subgroup of

*G*of order

*m*, then the number

*n/m*is an integer (call it

*k*). Given this surprising relationship, one might wonder if we can somehow “divide” the group

*G*by the subgroup

*H*and get a group of order

*k*. The answer to that is no. But, if the subgroup

*H*is

*normal*in

*G*, the answer is sort of yes, although we don’t use the term “divide” (but we do use the related term

*quotient*). In this situation, we can define a group of order

*k*that arises from the situation of

*H*as a normal subgroup of

*G*. Its elements are the (left or right) cosets of

*H*in

*G*, and it is called the

*quotient group G/H*(

*G*modulo

*H*). We have already kind of seen it in our discussion in Week 9 of how the image of a group homomorphism relates to the domain and the kernel. Relevant sections of the lecture notes this week are Section 4.2 and Section 4.3.

Slides for this week’s lectures.

Here are (old) video versions of this week’s lectures.