Characteristic submonoid of group not implies subgroup

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Statement

It is possible to have a group G and a submonoid M of G such that \sigma(M) = M for every automorphism \sigma of G such that M is not a subgroup of M. (By submonoid, we mean a subset that is a monoid with the binary operation and identity element inherited from the whole group).

Related facts

Similar facts

Opposite facts

Facts used

  1. Characteristically simple and abelian implies characteristic in holomorph: If A is a characteristically simple Abelian group, then A is characteristic in its holomorph: the semidirect product of A and \operatorname{Aut}(A).
  2. Automorphism group action lemma for quotients: Suppose A is an abelian normal subgroup of a group G, and \sigma is an automorphism of G that restricts to an automorphism \alpha of A and descends to an automorphism \sigma' of G/A. Then, if \rho:G/A \to \operatorname{Aut}(A) denotes the map by the conjugation action of the quotient (note: this requires abelianness of the subgroup), then \rho \circ \sigma' = c_\alpha \circ \rho.

Proof

Example involving the affine group

Consider the group G = GA(1,\mathbb{Q}) (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 x \mapsto ax + b from \mathbb{Q} to itself, with a \in \mathbb{Q} \setminus 0 and b \in \mathbb{Q}.
  • It is the group of upper triangular 2 \times 2 invertible matrices over the rationals, where both diagonal entries are equal.

Define A 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: x \mapsto x+ b.
  • It is the group of upper triangular 2 \times 2 matrices with 1s on the diagonal.

A lemma=

Given: G = GA(1,\mathbb{Q}), A is described as above, \sigma is an automorphism of G.

To prove: \sigma sends every coset of A in G to itself.

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 A is a characteristic subgroup of G. Fact (1), and that the group of rationals is characteristically simple Description of A in G. We can combine the given facts. Alternatively, we can explicitly compute that A is the derived subgroup of G.
2 \sigma restricts to an automorphism (say) \alpha of A and hence descends to an automorphism (say) \sigma' of G/A. Step (1) direct from Step (1).
3 \rho \circ \sigma' = c_\alpha \circ \rho where \rho:G/A \to \operatorname{Aut}(A) is the map given by the action of G on A by conjugation (see quotient group acts on abelian normal subgroup), and where c_\alpha denotes conjugation by \alpha. Fact (2) Step (2) Step-fact combination dorect.
4 \rho:G/A \to \operatorname{Aut}(A) is an isomorphism and that \operatorname{Aut}(A) is abelian The automorphism group of \mathbb{Q} is \mathbb{Q}^*.
5 \sigma' is the identity map on G/A. Steps (3), (4) By the abelianness of \operatorname{Aut}(A) in Step (4), c_\alpha is the identity map of \operatorname{Aut}(A), so plugging into Step (3), \rho \circ \sigma' = \rho. Since \rho is an isomorphism, \sigma' is the identity map.
6 \sigma sends every coset of A in G to itself. Steps (2), (5) This follows directly from Step (5) and by revisiting the definition of \sigma' in terms of \sigma.

An immediate corollary of the above is that every union of cosets of A in G is a characteristic subset of G.

Based on the above, we can construct many examples of characteristic submonoids of G that are not subgroups. The idea is to choose any submonoid of \mathbb{Q}^* \cong G/A, then take its inverse image in G under the quotient map to get the characteristic submonoid of G. Some examples are below.

Contraction mapping submonoid

Define M as the following submonoid (described in all the alternate descriptions):

  • It is the subset of G comprising those elements whose multiplicative group coordinate has modulus at most 1.
  • It is the set of all linear maps of the form x \mapsto ax + b where 0 < |a| \le 1 and a,b \in \mathbb{Q}.(in other words, it is those linear maps that are contractions).
  • It is the set of all upper triangular 2 \times 2 invertible matrices over the rationals where the two diagonal entries are equal and have absolute value at most 1.

Clearly, M is a submonoid of G -- this follows from the fact that the set of a for which 0 < |a| \le 1 form a submonoid of \mathbb{Q}^* under multiplication. On the other hand, M is not a subgroup of G. By the preceding observations, M is a characteristic submonoid of G that is not a subgroup.

Nonnegative powers of 2 mapping submonoid

  • It is the subset of G 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 x \mapsto 2^nx + b where n is a nonnegative integer.
  • It is the set of all upper triangular 2 \times 2 invertible matrices over the rationals where the two diagonal entries are equal to each other and their value is of the form 2^n, n a nonnegative integer.

Clearly, M is a submonoid of G -- this follows from the fact that the set of nonnegative integer powers of 2 is a submonoid of \mathbb{Q}^*. On the other hand, M is not a subgroup of G. By the preceding observations, M is a characteristic submonoid of G 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.