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I have encountered a few problems regarding the minimal subgroups of a finite group $G$. Any references and/or answers regarding the following questions will be very welcome.

1)If $G$ is a finite group, what can be said about its minimal normal subgroups (under inclusion)?

2) Is there a criterion to decide when an element $g \in G$ belongs to such a minimal subgroup?

3) Is the product of all minimal normal subgroups of a given finite group $G$ the entire group $G$?

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I presume "minimal normal subgroups" exclude the identity subgroup. For (3) you might ponder the symmetric groups. – Robin Chapman Jul 16 '10 at 10:47
Here #3 is obviously not the case, in the cyclic group of order 4. – Charles Matthews Jul 16 '10 at 10:48
The product of all minimal normal subgroups is stable under any automorphism of $G$. It is therefore a subgroup of Frattini's subgroup of $G$. – unknown Jul 16 '10 at 10:55
@unknown - care to explain how that works in a simple group? – Steve D Jul 21 '10 at 16:37
up vote 5 down vote accepted

Try reading the article on the socle of a group.

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Yup, that's what I needed. Thank you very much! I suppose that the proof of the fact that "Any finite characteristically simple group is the direct product of several copies of isomorphic simple groups" is not difficult. – unknown Jul 16 '10 at 11:04… – Charles Matthews Jul 16 '10 at 11:30
Thanks! Second question still remains unanswered. – unknown Jul 16 '10 at 11:50
I found a lot of theory relating the socle to permutation representations. E.g. . – Charles Matthews Jul 16 '10 at 13:01

Here is a little survey of the "bottom" of finite groups. It includes an answer to (2) in the case of finite nilpotent groups, and a correction to an anonymous comment.

The subgroup generated by the minimal normal subgroups is called the socle of the finite group. It is a direct product A×S where A is elementary abelian and S is a direct product of (non-abelian) simple groups. If A=1, then the group is a subgroup of Aut(S), and so has a very restricted structure. In general, A and S are not enough to determine the structure of the group very precisely (for instance A=2 allows both cyclic groups of order 2^n and the insoluble groups SL(2,q) for odd primes q). However, in special cases they can exert more control.

The socle of a finite abelian group is the subgroup consisting of the identity and the elements of prime order. The socle of a finite p-group (or finite nilpotent group) is the subgroup consisting of the identity and the central elements of prime order. Therefore you have very clear answers for (2) in the case of finite nilpotent groups: central of order dividing a prime.

For finite solvable groups, things are a little more complicated. A minimal normal subgroup must be elementary abelian, and so if g is in Soc(G), then N, the normal subgroup generated by g, must be elementary abelian since N ≤ Soc(G), and Soc(G) is a (direct product of) elementary abelian group(s). In particular, g commutes with all of its conjugates and has order dividing a prime. As the non-abelian group of order 6 shows, g need not be central.

If g commutes with all of its conjugates and has prime power order, then it is contained in every Sylow p-subgroup, but it need not be in the socle: the dihedral group of order 8 is already a counterexample. This indicates a subgroup that is easier to handle, exerts more control, but of course is larger: the Fitting subgroup.

The Fitting subgroup F is the direct product of the p-cores, where the p-core is the intersection of all of the Sylow p-subgroups. The Fitting subgroup can be built from above, below, and by horizontal or vertical slices, so it is extremely commonly assumed to be known. It is the largest normal nilpotent subgroup, and so contains the A part, A ≤ F. In a solvable group G, one always has G/Z(F) ≤ Aut( F ), so that you can view the "top" part of G as acting on the "bottom" part.

Often you want to do this with the socle, and there are (complicated) techniques for making it work in some cases, but the socle fails horribly for nilpotent groups: How can you understand the dihedral group of order 8 by the way it acts on its center of order 2? It cannot act at all on the center, it fixes both points.

The product of the minimal normal subgroups is not usually contained within the Frattini subgroup. For instance this fails for elementary abelian groups and for S3. When a chief factor (minimal normal subgroup of a quotient group) of a finite solvable group is contained in the Frattini, then it is called a Frattini factor, otherwise it is called complemented, since the (quotient) group is a semi-direct product with normal kernel that chief factor, and complement a (non-normal) maximal subgroup.

For insoluble groups, the socle is not very useful in general, but a very similar subgroup exerts an enormous amount of control, the so called layer L generated by all subnormal quasi-simple (nonabelian) subgroups. L contains S, but may intersect A a little: Fit(L) = A ∩ L. The subgroup F* generated by F and L is called the generalized Fitting subgroup, and it is large enough to detect the "top" of the group: G/Z(F) ≤ Aut(F*).

The difference between L and S is very small conceptually: S is the subgroup generated by the subnormal simple (non-abelian) subgroups, and L is the subgroup generated by the subnormal quasi-simple (non-abelian) subgroups. A quasi-simple group X is one where X/Z(X) is simple, but X is not X/Z(X) × Z(X). Non-simple quasi-simple groups are one of those somewhat rare occasions when the socle is contained within the Frattini, and that is exactly why the socle escapes detection when viewing quasi-simple groups and p-groups as automorphism groups.

As a rule of thumb, starting with the socle you can make bigger groups by either:

  • going up: these are called primitive groups and are described in many textbooks (and of course in many of Alexander Hulpke's works), or
  • going down: these are called perfect groups or p-groups and are especially difficult when the minimal normal subgroups are contained in the Frattini (e.g. the quasi-simple groups or any p-group).

Primitive groups can be studied by the way the group acts on its socle in many cases, indeed it is common (in those cases) to view the socle as the set of points which the primitive permutation group is permuting. Perfect and p-groups suddenly have a new socle, so they are not commonly studied by looking at the new socle (at the bottom), but rather by looking at their tops. In particular, I think you'll find looking at the bottoms of p-groups much harder than looking at the tops.

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