Melnikov group

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In mathematics, the Melnikov group or Melnikov subgroup of a profinite group ${\displaystyle G}$, denoted ${\displaystyle M(G)}$, is the intersection of all maximal open normal subgroups of ${\displaystyle G}$. If ${\displaystyle G}$ is nontrivial, then ${\displaystyle M(G)}$ is a proper closed subgroup invariant under every continuous automorphism of ${\displaystyle G}$ (that is, it is topologically characteristic). Fried and Jarden use the term "Melnikov group", while later literature also uses "Melnikov subgroup".

The construction is a normal-subgroup analogue of the Frattini subgroup ${\displaystyle \Phi (G)}$. Whereas the Frattini subgroup is defined using all maximal open subgroups, the Melnikov group uses only maximal open normal subgroups. Equivalently, ${\displaystyle M(G)}$ is the intersection of the kernels of all continuous epimorphisms from ${\displaystyle G}$ onto finite simple groups. It therefore records the part of ${\displaystyle G}$ that is invisible in every finite simple quotient.

Definition and simple quotients

Throughout, homomorphisms of profinite groups are understood to be continuous. A proper open normal subgroup ${\displaystyle N}$ of ${\displaystyle G}$ is maximal open normal if there is no proper open normal subgroup of ${\displaystyle G}$ strictly between ${\displaystyle N}$ and ${\displaystyle G}$. The Melnikov group is

${\displaystyle M(G)=\bigcap \{N\trianglelefteq G:N{\text{ is maximal open normal in }}G\}.}$

If ${\displaystyle N}$ is maximal open normal, then ${\displaystyle G/N}$ is a finite simple group. Conversely, if ${\displaystyle S}$ is a finite simple group and ${\displaystyle \varphi :G\twoheadrightarrow S}$ is an epimorphism, then ${\displaystyle \ker \varphi }$ is maximal open normal. Thus

${\displaystyle M(G)=\bigcap \{\ker \varphi$ :\varphi :G\twoheadrightarrow S,\;S{\text{ is a finite simple group}}\}.}

For a finite simple group ${\displaystyle S}$, define

${\displaystyle M_{S}(G)=\bigcap \{N\trianglelefteq G:G/N\cong S\},}$

with ${\displaystyle M_{S}(G)=G}$ when ${\displaystyle G}$ has no quotient isomorphic to ${\displaystyle S}$. Then

${\displaystyle M(G)=\bigcap _{S}M_{S}(G).}$

For each ${\displaystyle S}$, the quotient ${\displaystyle G/M_{S}(G)}$ is a Cartesian power of ${\displaystyle S}$. More precisely,

${\displaystyle G/M(G)\cong \prod _{S}S^{r_{G}(S)},}$

where ${\displaystyle S}$ ranges over the isomorphism classes of finite simple groups and ${\displaystyle r_{G}(S)}$ is the ${\displaystyle S}$-rank of ${\displaystyle G}$, equivalently the cardinal number of copies of ${\displaystyle S}$ occurring in ${\displaystyle G/M_{S}(G)}$. In particular,

${\displaystyle M(G)=1}$ if and only if ${\displaystyle G}$ is a Cartesian product of finite simple groups.

This also characterizes ${\displaystyle G/M(G)}$ as the largest quotient of ${\displaystyle G}$ that is a Cartesian product of finite simple groups: every homomorphism from ${\displaystyle G}$ onto such a product factors through ${\displaystyle G/M(G)}$.

Basic properties

The Melnikov group has several useful functorial and normal-generation properties. Let ${\displaystyle G}$ and ${\displaystyle H}$ be profinite groups.

• If ${\displaystyle \alpha :G\twoheadrightarrow H}$ is an epimorphism, then ${\displaystyle \alpha (M(G))=M(H)}$.
• If ${\displaystyle N}$ is a closed normal subgroup of ${\displaystyle G}$, then ${\displaystyle M(N)\leq M(G)}$.
• If ${\displaystyle N}$ is a closed normal subgroup of ${\displaystyle G}$ and ${\displaystyle NM(G)=G}$, then ${\displaystyle N=G}$.

The last property expresses the role of ${\displaystyle M(G)}$ in normal generation: adjoining ${\displaystyle M(G)}$ to a proper closed normal subgroup cannot make it equal to the whole group.

Relation with the Frattini subgroup

For every profinite group ${\displaystyle G}$, the Frattini subgroup is contained in the Melnikov group:

${\displaystyle \Phi (G)\leq M(G).}$

The containment can be strict. For the symmetric group ${\displaystyle S_{3}}$, viewed as a finite profinite group, the only maximal proper normal subgroup is ${\displaystyle A_{3}}$, so

${\displaystyle M(S_{3})=A_{3},}$

whereas ${\displaystyle \Phi (S_{3})=1}$.

For pro-${\displaystyle p}$ groups, however, the two constructions agree. Every maximal open subgroup of a pro-${\displaystyle p}$ group is normal and has index ${\displaystyle p}$, and consequently

${\displaystyle M(G)=\Phi (G)}$

for every pro-${\displaystyle p}$ group ${\displaystyle G}$.

Melnikov covers

An epimorphism ${\displaystyle \alpha :G\twoheadrightarrow H}$ of profinite groups is called a Melnikov cover if

${\displaystyle \ker \alpha \leq M(G).}$

This condition has a normal-generation interpretation: ${\displaystyle \alpha }$ is a Melnikov cover if and only if every closed normal subgroup ${\displaystyle N\trianglelefteq G}$ satisfying ${\displaystyle \alpha (N)=H}$ is equal to ${\displaystyle G}$. Melnikov covers are closed under composition. Moreover, a Melnikov cover induces an isomorphism

${\displaystyle G/M(G)\;\xrightarrow {\ \cong \ } \;H/M(H).}$

Thus a Melnikov cover preserves the quotient obtained after factoring out the Melnikov subgroup.

Uses

The Melnikov subgroup occurs in constructions that separate abelian and simple quotients of profinite groups. Bary-Soroker, Fehm and Wiese define a generalized derived subgroup ${\displaystyle D(G)}$ satisfying

${\displaystyle D(G)=M(G)\cap G',}$

where ${\displaystyle G'}$ is the commutator subgroup. Iterating this operation gives their abelian-simple length, which they use in criteria for the preservation of the Hilbertian property in algebraic extensions.

Also, this subgroup appears in modern structure theory for profinite groups

See also

• Frattini subgroup
• Profinite group
• Pro-p group
• Simple group

References

Bibliography