Subgroup-quotient duality for groups
There are many loose forms of subgroup-quotient duality, i.e., ways of transforming a definition or fact that is about subgroups to a similar fact about quotients, or more generally, ways of interchanging the roles played by subgroups and quotients in an existing definition. One way of thinking about this is that we are trying to "reverse the arrows" or go to the "opposite category" in a crude sense.
The quick glossary of this duality is given below:
Quick correspondence | Caveats and explanation |
---|---|
subgroup quotient group or normal subgroup quotient group |
The idea is that instead of quantifying statements over subgroups, we quantify them over quotient groups. |
injectivity surjectivity | The idea here is that the way injectivity interacts with subgroups is similar to the way surjectivity interacts with quotients. Similarly, the way injectivity interacts with quotients is similar to the way surjectivity interacts with subgroups. For instance, an injective map (not necessarily an endomorphism) from a group to itself restricts to a injective map on any subgroup to which its restriction is well-defined. However, this is not necessary for surjective maps. On the other hand, a surjective map, whenever it descends to a quotient, descends to a surjective map on the quotient. |
image of internal map (not necessarily homomorphism) kernel of internal map | We will see this more clearly explained in the verbal-marginal duality case. |
Contents
Situations where the duality is rigorous
The main situation where the duality is rigorous is the case of a finite abelian group. See subgroup lattice and quotient lattice of finite abelian group are isomorphic.
Subgroup properties
Below are listed some very basic subgroup properties along with their duals:
Subgroup property | Meaning | Dual property is the property of being a quotient group by what kind of subgroup? | Meaning |
---|---|---|---|
normal subgroup | invariant under all inner automorphisms | normal subgroup | the quotient group is invariant under inner automorphisms. |
characteristic subgroup | invariant under all automorphisms | characteristic subgroup | the quotient group is invariant under all automorphisms. |
injective endomorphism-invariant subgroup | invariant under all injective endomorphisms | strictly characteristic subgroup | the subgroup is invariant under all surjective endomorphisms, so these descend to the quotient group. |
fully invariant subgroup | invariant under all endomorphisms | fully invariant subgroup | invariant under all endomorphisms |
isomorph-free subgroup | no other isomorphic subgroups | quotient-isomorph-free subgroup | no other subgroup with isomorphic quotient group. |
isomorph-containing subgroup | any isomorphic subgroup is contained in it. | quotient-isomorph-containing subgroup | contained in any subgroup with an isomorphic quotient group. |
homomorph-containing subgroup | contains any homomorphic image of itself. | quotient-homomorph-containing subgroup | any homomorphic image of the quotient group is the quotient group by a subgroup containing it. |
verbal subgroup | image of a word map | marginal subgroup | elements that don't affect the word map (sort of like the kernel). |
direct factor | factor in an internal direct product | direct factor | factor in an internal direct product |
retract | subgroup with a normal complement | complemented normal subgroup | normal subgroup with a permutable complement. |
Basic verbal-marginal duality
The verbal-marginal duality is a loose, not very precise duality between statements that can be made about verbal subgroups and their factor groups versus marginal subgroups and their factor groups. The duality is difficult to work with rigorously, but it helps as an intuitive guide. The rough idea is that verbal subgroups start constructively by looking at images of word maps, so that the quotient groups are nice and "small" while marginal subgroups start by solving equations, so it is the subgroups themselves that are nice and "small."
Parameter/specification | Notion on the verbal side | Notion on the marginal side | Explanation of duality |
---|---|---|---|
a collection of words | verbal subgroup for the collection of words | the quotient group corresponding to the marginal subgroup for the collection of words. | the verbal subgroup is the image of the word map. The marginal subgroup is like the coordinate-wise kernel of the word map, so the quotient group by it should be qualitatively similar. Combining these pieces of data gives a notion of isologism of groups. |
a collection of words | quotient group by the verbal subgroup for the collection of words | the marginal subgroup for the collection of words | both these groups are in the variety of groups where the words are true. The quotient group is the largest such quotient. The subgroup is not the largest such subgroup, however. |
-- | derived subgroup | inner automorphism group | if you take the collection of all words obtained by taking products of commutators, the derived subgroup is the corresponding verbal subgroup and the inner automorphism group is the quotient group by the corresponding marginal subgroup. Combining these pieces of data gives a notion of isoclinism of groups. |
-- | abelianization | center | if you take the collection of all words obtained by taking products of commutators, the abelianization is the verbal quotient group and the center is the marginal subgroup. |
Group properties
Property | Meaning | Dual property | Meaning |
---|---|---|---|
perfect group | equals its own derived subgroup abelianization is trivial |
centerless group | the center is trivial, so the inner automorphism group is the whole group. |
Schur-trivial group | group in which every automorphism is inner | ||
superperfect group | Schur-trivial and perfect | complete group | centerless and every automorphism is inner. |
Group metaproperties
Property | Meaning | Dual property | Meaning |
---|---|---|---|
subgroup-closed group property | if a group has the property, so does every subgroup. | quotient-closed group property | if a group has the property, so does every quotient group. |
normal subgroup-closed group property | if a group has the property, so does every normal subgroup. | quotient-closed group property | if a group has the property, so does every quotient group. |
characteristic subgroup-closed group property | if a group has the property, so does every characteristic subgroup | characteristic quotient-closed group property | if a group has the property, so does every quotient group by a characteristic subgroup. |
Subgroup metaproperties
Metaproperty | Meaning: A subgroup property satisfies this metaproperty if ... | Dual metaproperty (translated into subgroup property terms from quotient property terms) | Meaning: A subgroup property satisfies this metaproperty if ... |
---|---|---|---|
transitive subgroup property | whenever are such that satisfies in and satisfies in , then satisfies in . | quotient-transitive subgroup property | whenever are such that is normal and satisfies in and further, satisfies in , then satisfies in . |
intermediate subgroup condition | whenever are such that satisfies in , then also satisfies in . | image condition | whenever is such that satisfies in , and is a surjective homomorphism, satisfies in . |
trivially true subgroup property | in any group , the trivial subgroup satisfies . | identity-true subgroup property | in any group , satisfies as a subgroup of itself. |