Proving that a subgroup is normal
This survey article is about proof techniques for or related to satisfaction of the following property: normal subgroup
Find other survey articles about normal subgroup | Find fact articles that prove satisfaction of this property
This article explores the various ways in which, given a group and a subgroup (through some kind of description) we can try proving that the subgroup is normal (or that it is not normal). We first discuss the leading general ideas, and then plunge into the specific cases.
Contents
- 1 Other things instead of proving normality
- 2 Using the standard definitions
- 3 Methods involving metaproperties of normality
- 4 Subgroup-defining function
- 5 Starting from normal subgroups and using deterministic processes
- 6 The deviation method of proving normality
- 7 Methods suited for particular groups
Other things instead of proving normality
In some cases, proving that a certain subgroup is normal may be hard or impossible, perhaps because the subgroup is not normal. The following alternative approaches are useful here:
- Replacing a subgroup by a normal subgroup: There are many techniques to guarantee, from the existence of a subgroup satisfying certain conditions, the existence of a normal subgroup satisfying similar conditions.
Using the standard definitions
The best way to try proving that a subgroup is normal is to show that it satisfies one of the standard equivalent definitions of normality.
Construct a homomorphism having it as kernel
This method is existential, or demonstrative -- exhibiting one homomorphism suffices.
To prove that is a normal subgroup of , we can construct a homomorphism such that the kernel of the homomorphism, i.e., the set of elements that map to the identity, is precisely . Note that if we do this successfully, it is not even necessary to establish separately that is a subgroup.
Here are some examples:
Statement | Statement with symbols | Type | Quick explanation of proof using this approach |
---|---|---|---|
Every group is normal in itself (proof section) | For any group , is normal in | Subgroup metaproperty satisfaction for identity-true subgroup property | Consider the trivial homomorphism from to the trivial group, i.e., the homomorphism that sends every element of to the identity element of the trivial group. The kernel of this is itself. |
Trivial subgroup is normal | In any group , the trivial subgroup (the subgroup comprising only the identity element) is normal | Subgroup metaproperty satisfaction for trivially true subgroup property | Consider the identity map from to itself. This is a homomorphism, and its kernel is the trivial subgroup. |
Normality is strongly intersection-closed | If are all normal in , so is | Subgroup metaproperty satisfaction for strongly intersection-closed subgroup property | Consider the homomorphism from to the external direct product of all the quotient groups , which in each coordinate is the corresponding quotient map. The kernel of this is the intersection . |
Normality satisfies intermediate subgroup condition | , normal in , then normal in | Subgroup metaproperty satisfaction for intermediate subgroup condition | Compose the inclusion of in with the quotient map from to . This is a homomorphism from with kernel . |
Center is normal | The center is normal in | Subgroup-defining function property satisfaction for center | Center is the kernel of the homomorphism from the group to its automorphism group via the conjugation action. |
Direct factor implies normal | , then is normal in . | Subgroup property implication from direct factor | is the kernel of the projection to the second coordinate: . |
Verify invariance under inner automorphisms
This method requires a universal check.
Another way of proving normality is using the inner automorphism, or conjugation, definition. This states that a subgroup of a group is normal if for every , we have .
This definition could be used in three broad ways:
- Try everything: Here, we basically check every element of and every element of .
- Use generic elements: Here, we don't actually try each element, but rather, argue that for an arbitrary choice of element of and element of , the result holds.
- Use generating sets: This is a somewhat stronger version. It says that if is a generating set for and is a generating set for , then is normal in if for every , and are both in .
The generating set approach is also useful in computational situations, both when dealing with groups described by presentations and with groups described by means of generating sets inside larger ambient groups. For more on this, see normality testing problem.
Here are some applications of the generic element approach:
Statement | Statement with symbols | Type | Quick explanation of proof using this approach |
---|---|---|---|
Every group is normal in itself (proof section) | For any group , is normal in | Subgroup metaproperty satisfaction for identity-true subgroup property | Boils down to closure of the group under multiplication, inverse map |
Trivial subgroup is normal | In any group , the trivial subgroup (the subgroup comprising only the identity element) is normal | Subgroup metaproperty satisfaction for trivially true subgroup property | Boils down to using the property of identity element and inverses to show |
Normality is strongly intersection-closed | If are all normal in , so is | Subgroup metaproperty satisfaction for strongly intersection-closed subgroup property | For , for each , hence in intersection. |
Normality satisfies intermediate subgroup condition | , normal in , then normal in | Subgroup metaproperty satisfaction for intermediate subgroup condition | use the fact that any element of is also in . |
Center is normal | The center is normal in | Subgroup-defining function property satisfaction for center | In fact, for , , . |
The generating set approach, or ideas of that kind, are more useful when the subgroup is described by means of generating elements or as a join of subgroups. For instance:
Statement | Statement with symbols | Type | Quick explanation of proof using this approach |
---|---|---|---|
Normality is strongly join-closed | normal in , then so is | Subgroup metaproperty satisfaction | Any element of the join is a product of elements in the s. Use the fact that conjugation is an automorphism and invariance of each letter in the product. See also endo-invariance implies strongly join-closed |
Normal subset generates normal subgroup | If is a normal subset of , is a normal subgroup | Random fact | Write arbitrary element of as a product of elements of and their inverses, now use that conjugation is an automorphism. |
Determine its left and right cosets
This method requires a universal check.
A subgroup is normal in a group iff, for every , . This definition is useful for proving normality is some situations:
Statement | Statement with symbols | Type | Quick explanation of proof using this approach |
---|---|---|---|
Every group is normal in itself (proof section) | For any group , is normal in | Subgroup metaproperty satisfaction for identity-true subgroup property | For any , and . So . |
Trivial subgroup is normal | In any group , the trivial subgroup (the subgroup comprising only the identity element) is normal | Subgroup metaproperty satisfaction for trivially true subgroup property | For any , and . So . |
Subgroup of index two is normal | If the index of a subgroup in is 2, then is normal in | Subgroup property implication from subgroup of index two | For any , . For any , is the complement of in , and so is . |
Center is normal | The center is normal in | Subgroup-defining function property satisfaction for center | In fact, for , , . Thus, element-wise and hence also as whole sets. |
Compute its commutator with the whole group
A subgroup of a group is normal iff the commutator is contained in . This definition is useful for proving, for instance, that the derived subgroup is normal.
Methods involving metaproperties of normality
Joins and intersections
Further information: Normality is strongly join-closed,Normality is strongly intersection-closed
If the given subgroup can be described using joins and intersections ,starting with normal subgroups, then it is normal.
Upper joins
Further information: Normality is upper join-closed
If the given subgroup is normal in a bunch of intermediate subgroups that together generate the whole group, it is normal in the whole group.
Quotient-transitivity
Further information: Normality is quotient-transitive
If are such that is normal in and is normal in , then is normal in . This is a frequently used fact.
Images, inverse images, transfer, intermediate subgroups
Further information: Normality satisfies intermediate subgroup condition, Normality satisfies transfer condition, Normality satisfies image condition, Normality satisfies inverse image condition
Centralizers
Further information: Normality is centralizer-closed
The centralizer of a normal subgroup is normal. Thus, a group obtained by starting from a normal subgroup and taking the centralizer is normal. Moreover, the taking the centralizer operation can be combined with joins, intersections, and many other operations.
Commutators
Further information: Normality is commutator-closed, commutator of a group and a subgroup implies normal, commutator of a group and a subset implies normal
The commutator of two normal subgroups is also a normal subgroup. Also, the commutator of the whole group and any subset is normal. These facts can be useful in establishing that certain subgroups are normal.
Subgroup-defining function
One of the simplest ways of showing that a subgroup is normal is to show that it arises from a subgroup-defining function. A subgroup-defining function is a rule that associates a unique subgroup to the group.
Any subgroup obtained via a subgroup-defining function is invariant under any automorphism of the group, and is hence a characteristic subgroup. In particular, it is invariant under inner automorphisms of the group, and is hence normal.
With this approach, for instance, we can show that the center, the derived subgroup, and the Frattini subgroup are normal.
Starting from normal subgroups and using deterministic processes
An even more general idea than that of subgroup-defining functions is the following: any subgroup that is obtained by starting from a collection of normal subgroups and using deterministic processes, which may involve joins, intersections, centralizers, and commutators, or other processes, still yields a normal subgroup. Here, the term deterministic means invariant under automorphisms of the entire system, which implies, in particular, invariance under inner automorphisms.
The deviation method of proving normality
In measuring deviation from normality, we see three ways of measuring the extent to which a subgroup deviates from normality: the normalizer, the normal closure and the normal core. Here, we explore each of these as a tool for trying to prove normality.
The normal core method and group actions
Further information: Group acts on left coset space of subgroup by left multiplication
The idea behind using the normal core to prove normality is to show that the given subgroup equals its normal core: the largest normal subgroup contained in it. In other words, we try to establish that the intersection of all conjugates of the subgroup equals the subgroup itself. This method is particularly useful in cases where the subgroup has small index in the whole group.
The normal core method is typically applied along with group actions, as in the setup described below.
Let be a subgroup of . Then, acts on the coset space of . This gives a homomorphism from to the symmetric group on the coset space, and the kernel of the homomorphism is the normal core . Hence, the quotient group sits as a subgroup of the symmetric group .
This approach can be used to prove results like the following:
- Index two implies normal
- Subgroup of least prime index is normal
- A related result, that does not prove the subgroup itself is normal, but that it contains another large enough normal subgroup: Poincare's theorem, which states that any subgroup of index contains a normal subgroup of index dividing .
Normal closure
The idea behind using the normal closure in order to prove normality is to prove that the subgroup equals its own normal closure. In other words, we show that the subgroup equals that subgroup generated by all its conjugates. This method is particularly useful when the subgroup is given in terms of a generating set.
Suppose is a group and is a subgroup with generating set . The normal closure of in can be obtained as the subgroup generated by all conjugates of elements of be elements of . Thus, to show that is normal in , it suffices to show that all such conjugates are again in .
In fact, if we are given a generating set for , it suffices to prove that conjugating any element of by any element in gives an element of .
Normalizer
The idea behind using the normalizer to prove normality is to prove that the normalizer of the subgroup equals the whole group. In other words, we show that every element of the group commutes with the subgroup.
Methods suited for particular groups
For abelian groups
If the whole group is abelian, then every subgroup is normal, so there is nothing to prove. Further information: Abelian implies every subgroup is normal