Varying simplicity

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This is a survey article describing variations on the following: simple group
View other variational survey articles | View other survey articles about simple group

Simplicity is one of the most pivotal group properties. Simple groups play a role in group theory analogous to the role primes play in number theory, except that the structure of simple groups is far more complicated than that of primes, and further, the ways simple groups can be assembled are far more diverse than the way primes can be multiplied.

The article looks at different directions in which the notion of simple group can be altered, modified and stretched. Most of the changes are in the weakening direction, viz they give properties that are satisfied by all the simple groups, as well as by some other groups.

Variations that employ the simple group operator

The simple group operator takes as input a trim subgroup property (viz a subgroup property satisfied by both the trivial subgroup and the whole group) and outputs the property of being a group which has no proper nontrivial subgroup with that property.

The property of being a simple group arises by applying the simple group operator to the subgroup property of being normal.

Thus, to get variations of the notion of simple group, we apply the simple group operator to subgroup proeprties that are themselves obtained by varying normality.

Characteristically simple group

Further information: characteristically simple group

The group property of being characteristically simple is obtained by applying the simple group operator to the subgroup property of being characteristic.

In other words, a group is characteristically simple if it has no proper nontrivial characteristic subgroup. It turns out that this is equivalent to demanding that the group be a direct product of isomorphic simple groups.

Any minimal normal subgroup of a group is characteristically simple.

Directly indecomposable group

Further information: directly indecomposable group

The most direct analogue of factorizing natural numbers for the case of groups, is factoring a group as a direct product of subgroups. A directly indecomposable group is a group that does not possess any nontrivial factorization of this sort.

More formally, the group property of being directly indecomposable is obtained by applying the simple group operator to the subgroup property of being a direct factor, viz a group is directly indecomposable if and only if it has no proper nontrivial direct factor.

Splitting-simple group

Further information: splitting-simple group

A group is termed splitting-simple, also semidirectly indecomposable, it it cannot be expressed as an internal semidirect product of a nontrivial normal subgroup and a nontrivial subgroup. In other words, the property of being splitting-simple is obtained by applying the simple group operator to the property of being a complemented normal subgroup. It's also obtained by applying the simple group operator to the property of being a retract (a subgroup possessing a normal complement).

Centrally indecomposable group

Further information: centrally indecomposable group

A group is termed centrally indecomposable if it cannot be expressed as the central product of two proper subgroups. Note that for non-abelian groups, this is equivalent to sayin that the group has no proper nontrivial central factor.

Strictly simple group

Further information: strictly simple group

A group is termed strictly simple if it possesses no proper nontrivial ascendant subgroup. This property is the same as simplicity for finite groups, and indeed, for slender groups (groups with an ascending chain condition on subgroups).

Absolutely simple group

Further information: absolutely simple group

A group is termed absolutely simple if it possesses no proper nontrivial serial subgroup. This property is the same as simplicity for finite groups, and indeed, for slender groups.

Primitivity and group action-related ideas

Primitive group

Further information: primitive group

In a simple group, every subgroup is core-free. A primitive group is a somewhat weaker notion: a group is primitive if it has a maximal subgroup which is core-free.

This essentially means that there is a maximal subgroup such that the action of the group on the coset space of that subgroup is trivial. (The fact of the subgroup in question being maximal is characterized by saying that the action on the coset space is a primitive group action).

Primitive groups play a role in permutation group theory similar to the role that simple groups play -- we can somehow look at a group action and obtain a structure forest for it which breaks it up into primitive group actions.

Other related notions

Via the group property core operator

The group property core operator takes a normal join-closed group property and outputs a subgroup-defining function that takes in a group and gives out the unique largest normal subgroup of the group satisfying that property.

Note that if we take a simple group, then any group property core operator will output, for the given group, either the trivial subgroup or the whole group. In many cases, we can show that the only simple groups for which the core is the whole group are the simple Abelian groups (which are very few). Thus, the property of having a trivial core is a weakening of the property of being a simple non-Abelian group.

Fitting-free group

Further information: Fitting-free group

A group is said to be Fitting-free if its Fitting subgroup is trivial, viz., it has no nontrivial normal nilpotent subgroup. Equivalently, a group is said to be Fitting-free if it has no nontrivial normal Abelian subgroup, or it has no proper nontrivial normal solvable subgroup.

Core-free group

Further information: core-free group

A finite group is said to be core-free if its Brauer core is trivial, viz., it has no nontrivial normal subgroup of odd order.

Miscellaneous variations involving automorphisms and inner automorphisms

Almost simple group

Further information: Almost simple group

Note that any simple non-Abelian group is centerless, viz, its center is trivial. Hence, the natural map to the automorphism group, that sends each element to the conjugation induced by it, is injective. So a simple non-Abelian group embeds as a subgroup of its automorphism group.

An almost simple group is a group that occurs as some intermediate subgroup between a simple non-Abelian group and its automorphism group.

Quasisimple group

Further information: Quasisimple group

Note that any simple non-Abelian group is perfect, viz., its commutator subgroup is the whole group. Further, it equals its own inner automorphism group, and hence the inner automorphism group is simple.

A quasisimple group is a group which satisfies both the conditions, viz., being perfect and having a simple inner automorphism group.

Group in which every endomorphism is trivial or an automorphism

Further information: Group in which every endomorphism is trivial or an automorphism

Any finite simple group has the property that any endomorphism of it is either trivial or an automorphism. More generally, any simple co-Hopfian group has the property that every endomorphism is either trivial or an automorphism. Further, any finite quasisimple group also has the property that every endomorphism is trivial or an automorphism.

Thus, there seems to be a relation between simplicity and there not existing endomorphisms other than automorphisms and the trivial map.