5.3: Normal Subgroups
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We’ve seen an example where the left and right cosets of a subgroup were different and a few examples where they coincided. In the latter case, the subgroup has a special name.
Let \(G\) be a group and let \(H\leq G\). If \(aH=Ha\) for all \(a\in G\), then we say that \(H\) is a normal subgroup. If \(H\) is a normal subgroup of \(G\), then we write \(H\trianglelefteq G\).
Provide an example of group that has a subgroup that is not normal.
There are a few instances where we can guarantee that a subgroup will be normal.
Suppose \(G\) is a group. Then \(\{e\}\trianglelefteq G\) and \(G\trianglelefteq G\).
If \(G\) is an abelian group, then all subgroups of \(G\) are normal.
A group does not have to be abelian in order for all the proper subgroups to be normal.
Argue that all of the proper subgroups of \(Q_8\) are normal in \(Q_8\).
Suppose \(G\) is a group and let \(H\leq G\) such that \([G:H]=2\). Then \(H\trianglelefteq G\).
It turns out that normality is not transitive.
Consider \(\langle s\rangle=\{e,s\}\) and \(\langle r^2,sr^2\rangle =\{e,r^2,sr^2,s\}\). It is clear that \[\langle s\rangle\leq \langle r^2,sr^2\rangle\leq D_4.\] Show that \(\langle s\rangle\trianglelefteq \langle r^2,sr^2\rangle\) and \(\langle r^2,sr^2\rangle\trianglelefteq D_4\), but \(\langle s\rangle\not\trianglelefteq D_4\).
The previous problem illustrates that \(H\trianglelefteq K \trianglelefteq G\) does not imply \(H\trianglelefteq G\).
Suppose \(G\) is a group and let \(H\leq G\). For \(g\in G\), we define the conjugate of \(H\) by \(g\) to be the set \[gHg^{-1}:=\{ghg^{-1}\mid h\in H\}.\]
Suppose \(G\) is a group and let \(H\leq G\). Then \(H\trianglelefteq G\) if and only if \(gHg^{-1}\subseteq H\) for all \(g\in G\).
Another way of thinking about normal subgroups is that they are “closed under conjugation." It’s not too hard to show that if \(gHg^{-1}\subseteq H\) for all \(g\in G\), then we actually have \(gHg^{-1}=H\) for all \(g\in G\). This implies that \(H\trianglelefteq G\) if and only if \(gHg^{-1}=H\) for all \(g\in G\). This seemingly stronger version of Theorem \(\PageIndex{4}\) is sometimes used as the definition of normal subgroup. This discussion motivates the following definition.
Let \(G\) be a group and let \(H\leq G\). The normalizer of \(H\) in \(G\) is defined via \[N_G(H):=\{g\in G\mid gHg^{-1}=H\}.\]
If \(G\) is a group and \(H\leq G\), then \(N_G(H)\) is a subgroup of \(G\).
If \(G\) is a group and \(H\leq G\), then \(H\trianglelefteq N_G(H)\). Moreover, \(N_G(H)\) is the largest subgroup of \(G\) in which \(H\) is normal.
It is worth pointing out that the “smallest" \(N_G(H)\) can be is \(H\) itself—certainly a subgroup is a normal subgroup of itself. Also, the “largest" that \(N_G(H)\) can be is \(G\), which happens precisely when \(H\) is normal in \(G\).
Find \(N_{D_4}(V_4)\).
Find \(N_{D_3}(\langle s\rangle)\).
We conclude this chapter with a few remarks. We’ve seen examples of groups that have subgroups that are normal and subgroups that are not normal. In an abelian group, all the subgroups are normal. It turns out that there are examples of groups that have no normal subgroups. These groups are called simple groups. The smallest simple group is \(A_5\), which has 60 elements and lots of proper nontrivial subgroups, none of which are normal.
The classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories:
- A cyclic group with prime order;
- An alternating group of degree at least 5;
- A simple group of Lie type, including both
- the classical Lie groups, namely the simple groups related to the projective special linear, unitary, symplectic, or orthogonal transformations over a finite field;
- the exceptional and twisted groups of Lie type (including the Tits group);
- The 26 sporadic simple groups.
These groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the prime numbers are the basic building blocks of the natural numbers.
The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups (and their action on other mathematical objects) can sometimes be reduced to questions about finite simple groups. Thanks to the classification theorem, such questions can sometimes be answered by checking each family of simple groups and each sporadic group. The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.
The classification of the finite simple groups is a modern achievement in abstract algebra and I highly encourage you to go learn more about it. You might be especially interested in learning about one of the sporadic groups called the Monster Group.