Note to the reader: This article assumes basic knowledge of Groups, Subgroups and related concepts like Cyclic Groups, Cosets, modulo arithmetic etc. Due to the expository nature of this article, the reader might have previously encountered several ideas discussed here. The objective is to present the key ideas in a logical and lucid manner in order to make a good sense of the abstract concepts being discussed.
Abstract: This article aims to briefly summarize the motivation behind the concept of Normal subgroups, the equivalent definitions of a subgroup being Normal and its relation to a new group structure namely, the Quotient Group.
Motivation: We will begin by giving a motivation for the Quotient groups. In elementary school, when we had our first encounter with the ‘division’ operation on ‘numbers’, we were told that, a÷b means that the quantity a is divided in b equal ‘parts’. For e.g. if we have 18 oranges and need to equally distribute them in 2 people each person would get 9 oranges or if we were to distribute among 3 people each person would get 6 oranges. Intuitively, we are breaking the collection of 18 objects into of 2 sub-collections of 9 objects each and breaking the collection of 18 objects into 3 sub-collections of 6 objects each, respectively.
Just like the numbers, it turns out we can perform similar operations on abstract structures likes Groups, Rings, Vector spaces etc. too!
To understand this division scheme on groups, it is intuitively clear that we need to ‘divide’ a group by another group. Normally, a nice division in the numbers occurs when the divisor is smaller than the dividend and the dividend is a multiple of the divisor. Intuitively, we follow an analogous process by attempting to partition the group via one of its subgroups but not just any subgroup, just like an integer cannot be divided by all the integers to obtain an integer quotient not every group can be divided by just any of its subgroups to obtain a quotient which is also a group. We will now investigate the properties a subgroup must have in order to ‘divide’/ partition a group.
Introduction: We now begin with a more concrete example in a more formal setting. Consider the group of integers under usual addition, (Z,+). We know that every subgroup of Z is of the form nZ, where n is a natural number. Observe that we can partition the set Z via the left cosets of nZ. For e.g choose n=5. then we can partition Z in the following manner (refer to Fig. 2):
We begin by taking 5Z, then we take left cosets. Notice that each r in Fig. 2 gives the remainder of each number in a coset under division by 5. It is a good exercise to see that for any r>4 the cosets co-incides back to the one of the above 5 cosets. For e.g. 6+5Z = {…,-9, -4, 1, 6, 16,…} which is the same as 1+5Z or in general for any integer a, a+5Z is same as (a mod 5)+ 5Z. Clearly, Z is partioned now and the corresponding equivalence relation to the partition here is addition modulo 5! Now, if we treat the cosets as ‘numbers’ we get a group of 5 elements under the addition modulo 5 operation! This group is nothing but our famous Z/5Z group under addition modulo 5. Observe, 5Z or in general nZ partitions Z into 5 or n cosets respectively which is analogous to the division in integers! The group operation for the cosets is that left cosets are mapped to left cosets i.e. (a+nZ) + (b+nZ)=(a+b)+nZ and we have seen above that (a+b)+nZ is equivalent to {(a+b)mod 5}+nZ.
Exercise: Verify the cosets of nZ form a group of n elements, namely Z/nZ={nZ, 1+nZ, 2+nZ,…, (n-1)+nZ}.
Remark: The cosets after the partitioning are also viewed as equivalence classes and are denoted by [0], [1],.., [n-1].
Normal Subgroups: We can now safely move towards the generalization of the above example. In general, not every subgroup exhibits the power to partition the group via its cosets and form a new group of cosets. To investigate this, let N be a subgroup of some group G. Pick any two left cosets of N, xN, yN for any x, y in G. Clearly, e is in N. This means, x.e=x is in xN and similarly y is also in yN. We assume that the cosets of N form a group with the coset operation as defined earlier. Since the cosets form the group we have the following.
This gives us the property the subgroup N should have i.e. y^{-1}ny belongs to N. Such a subgroup is called a Normal subgroup. It is denoted as N◁G. Notice that every subgroup of an abelian group is normal. This follows directly by the commutativity:
This means that for every y and n y^{-1}ny belongs to N and hence any subgroup N of an abelian group is normal. Similarly, one can easily show that, every subgroup of any cyclic group is normal.
Remark: Cosets of any subgroup of a group partitions the group and one can always find the corresponding equivalence relation. The set of the equivalence classes is called the ‘Quotient Set’. However, not every quotient set qualifies to admit a group structure. Nice things begin to happen if the subgroup is normal.
Equivalent Definitions: We will now move one step further on finding out the equivalent conditions which make a subgroup normal.
The next equivalent definition follows immediately from our second definition which we just proved in Theorem 1.
The last definition tells us a very important fact about normal subgroups i.e. N commutes with every element of G. i.e. the left and right cosets are the same for a normal subgroup. We began from the left cosets and have shown that the left and the right cosets agree for a normal subgroup. Hence, we could have started our definitions with the right cosets, it does not matter!
We have now seen what a normal subgroup is. But we have missed an important question regarding the group of cosets. We started our argument by assuming the set of cosets to be a group and deduced the conditions to be imposed on the subgroup N i.e. If the cosets form a group then the subgroup is called normal. But does the converse hold? The answer is yes! A normal subgroup partitions a group G into a Quotient group. Note that the quotient group is NOT a subgroup of the group G(refer to the integer modulo example). We will now prove the converse.
Thus, a subgroup being normal automatically validates that it can factor the group into a new group of cosets called as the Quotient or Factor Group and the other way round. The examples in beginning make the reasoning behind the nomenclature ‘factor’ or ‘quotient’ group, pretty obvious . We would end our discussion by introducing the definition of a Simple Group. If the only normal subgroups of a group G are {e} and G itself then G is called a simple group. It is now very evident that like the prime integers simple groups can also be factored by only the trivial group i.e. the identity or the group itself. Simple groups, analogous to prime numbers form the building blocks of other groups, discussion of which is beyond the scope of this exposition. Hence, we now conclude our discussion on normal subgroups and quotient groups.