Edit: It may be a little opinion-based, so I’ve posted it on MathOverflow: The importance/use of socle in the theory of finite groups.
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Actually, I’m new here and not quite sure whether I should post this question on this site. If you think it is more suitable for MathOverflow, please tell me and I will delete this post.
Definition. The socle of a group $G$, denoted ${\rm Soc}(G)$, is the subgroup generated by the minimal normal subgroups of $G$.
Here we only discuss it in the theory of finite groups.
I know some basic facts.
- ${\rm Soc}(G)$ is the direct product of some of the minimal normal subgroups of $G$.
- ${\rm Soc}(G)$ is semisimple.
- ${\rm Soc}(H\times K)={\rm Soc}(H)\times {\rm Soc}(K)$.
- ${\rm Soc}({\rm Soc}(G))={\rm Soc}(G)$.
- If $G$ is nilpotent, then ${\rm Soc}(G)$ is central and hence abelian.
- ••••••
I can find the definition of socle in many, though not all, of the text books, but always not much is discussed about it. They don’t seem to attach much importance to the concept of socle.
I think the concept of socle is important, because it is literally the “socle”, the plinth, of a group.
So my question is: How is this concept used in the theory of finite groups? I’m a beginner and I want to know if there is any important use of the concept of solcle.
I’m interested in it. Is there any theorem or article or book that you think I should know and read about it?
Any comment or answer is welcome. Any help is sincerely appreciated. Thanks!
Best Answer
Maybe the socle itself isn't that important, but its constituent parts, the minimal normal subgroups, are very important, particularly in soluble groups.
Here is one vitally important example. Let $G$ be a primitive permutation group (i.e., transitive and the point stabilizer is a maximal subgroup). If $G$ is soluble, then the degree of $G$ is a power $p^n$ of a prime $p$. This degree is the order of some minimal normal subgroup, which since $G$ is soluble, must be an elementary abelian $p$-group (direct product of cyclic groups of order $p$). This subgroup is regular on the set.
The socle appears in primitive permutation groups, because of the O'Nan--Scott theorem, which sets out to list all primitive permutation groups in some sense (it originally classified the maximal subgroups of symmetric groups, but has been adapted). It's important to know whether there is more than one minimal normal subgroup (i.e., whether the socle is minimal normal or a product of two).
Another use of minimal normal subgroups is in reduction theorems for finite groups. Let $G$ be an arbitrary finite group, and suppose you want to prove some conjecture about it. You have the classification of finite simple groups, and you want to apply that. This means you need to reduce the problem to a question about simple groups.
For some problems it is often easy to eliminate the case where there is an abelian normal subgroup. Thus a minimal normal subgroup is a direct product of isomorphic simple groups. The socle of this group, often called the layer or the Bender subgroup in this situation, contains its own centralizer. (Note that this requires there to be no abelian normal subgroups!) If $N$ is a normal subgroup of $G$ such that $C_G(N)\leq N$ and $N$ is a product of non-abelian simple groups, then $C_G(N)=1$ and so $G$ is a subgroup of $\mathrm{Aut}(N)$.
Another example is with classifying maximal subgroups of finite groups. A theorem of Aschbacher and Scott states that one can understand the maximal subgroups of all finite groups if and only if one can understand the maximal subgroups of groups $M\rtimes G$, where $G$ is a simple group (technically, almost simple, but simple is interesting enough), and $M$ is an elementary abelian $p$-group for some prime $p$ on which $G$ acts. Here the socle is $M$, and this reduction theorem was obtained by looking precisely at this sort of minimal normal structure.