Characteristic submonoid of group not implies subgroup
It is possible to have a group and a submonoid of such that for every automorphism of such that is not a subgroup of . (By submonoid, we mean a subset that is a monoid with the binary operation and identity element inherited from the whole group).
- Nonempty finite subsemigroup of group is subgroup: Any finite nonempty subsemigroup of a group is a subgroup. More generally, for a periodic group, i.e., a group where every element has finite order, any nonempty subsemigroup is a subgroup.
- Nonempty characteristic subsemigroup of abelian group implies subgroup: For an abelian group, any characteristic submonoid, and more generally, any nonempty characteristic subsemigroup, must be a subgroup. This is because the inverse map is an automorphism for Abelian groups.
- Characteristically simple and abelian implies characteristic in holomorph: If is a characteristically simple Abelian group, then is characteristic in its holomorph: the semidirect product of and .
- Automorphism group action lemma for quotients: Suppose is an abelian normal subgroup of a group , and is an automorphism of that restricts to an automorphism of and descends to an automorphism of . Then, if denotes the map by the conjugation action of the quotient (note: this requires abelianness of the subgroup), then .
Example involving the affine group
Consider the group (see general affine group:GA(1,Q)): the general affine group over the rational numbers. this group can be described concretely in many ways:
- It is the semidirect product of the additive group of rational numbers by the multiplicative group (i.e., it is the holomorph of the additive group of rational numbers).
- It is the group (under composition) of all linear maps from to itself, with and .
- It is the group of upper triangular invertible matrices over the rationals, where both diagonal entries are equal.
Define as the following subgroup:
- It is the normal subgroup comprising the additive group of rational numbers: the base of the semidirect product.
- It is the subgroup comprising the translation maps: .
- It is the group of upper triangular matrices with s on the diagonal.
Given: , is described as above, is an automorphism of .
To prove: sends every coset of in to itself.
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||is a characteristic subgroup of .||Fact (1), and that the group of rationals is characteristically simple||Description of in .||We can combine the given facts. Alternatively, we can explicitly compute that is the derived subgroup of .|
|2||restricts to an automorphism (say) of and hence descends to an automorphism (say) of .||Step (1)||direct from Step (1).|
|3||where is the map given by the action of on by conjugation (see quotient group acts on abelian normal subgroup), and where denotes conjugation by .||Fact (2)||Step (2)||Step-fact combination dorect.|
|4||is an isomorphism and that is abelian||The automorphism group of is .|
|5||is the identity map on .||Steps (3), (4)||By the abelianness of in Step (4), is the identity map of , so plugging into Step (3), . Since is an isomorphism, is the identity map.|
|6||sends every coset of in to itself.||Steps (2), (5)||This follows directly from Step (5) and by revisiting the definition of in terms of .|
An immediate corollary of the above is that every union of cosets of in is a characteristic subset of .
Based on the above, we can construct many examples of characteristic submonoids of that are not subgroups. The idea is to choose any submonoid of , then take its inverse image in under the quotient map to get the characteristic submonoid of . Some examples are below.
Contraction mapping submonoid
Define as the following submonoid (described in all the alternate descriptions):
- It is the subset of comprising those elements whose multiplicative group coordinate has modulus at most .
- It is the set of all linear maps of the form where and .(in other words, it is those linear maps that are contractions).
- It is the set of all upper triangular invertible matrices over the rationals where the two diagonal entries are equal and have absolute value at most .
Clearly, is a submonoid of -- this follows from the fact that the set of for which form a submonoid of under multiplication. On the other hand, is not a subgroup of . By the preceding observations, is a characteristic submonoid of that is not a subgroup.
Nonnegative powers of 2 mapping submonoid
- It is the subset of comprising those elements whose multiplicative group coordinate is a nonnegative integer power of 2.
- It is the set of all linear maps of the form where is a nonnegative integer.
- It is the set of all upper triangular invertible matrices over the rationals where the two diagonal entries are equal to each other and their value is of the form , a nonnegative integer.
Clearly, is a submonoid of -- this follows from the fact that the set of nonnegative integer powers of 2 is a submonoid of . On the other hand, is not a subgroup of . By the preceding observations, is a characteristic submonoid of that is not a subgroup.
Nilpotent group example
Further information: characteristic submonoid of nilpotent group not implies subgroup
It is possible to construct a nilpotent group (in fact, a nilpotent group in which every automorphism is inner) and a characteristic submonoid of the group (in fact, one that is contained in the center) that is not a subgroup.