Frattiniembedded normalrealizable implies innerinautomorphismFrattini
From Groupprops
This fact is related to the problem of realization related to the following subgroupdefining function: Frattini subgroup
Realization problems are usually about which groups can be realized as subgroups/quotients related to a subgroupdefining function.
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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property must also satisfy the second group property
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Statement
Suppose is a Frattiniembedded normalrealizable group: in other words, can be embedded in some group as a Frattiniembedded normal subgroup. Then satisfies the following condition:
The inner automorphism group of is a Frattiniembedded normal subgroup of the automorphism group of .
Proof
Proof outline
Using notation as above, let be the group and be a group in which is embedded as a Frattiniembedded normal subgroup. Then:
 Frattiniembedded normal is quotientclosed: Consider the map from to , sending an element of to its action on by conjugation. Let be the image of . Using the fact that the property of being Frattiniembedded normal is preserved upon taking quotients, we see that is a Frattiniembedded normal subgroup of
 Frattiniembedded normal in subgroup and normal implies Frattiniembedded normal: is normal in , and is Frattiniembedded normal in the intermediate subgroup , so this result tells us that is Frattiniembedded normal in
In case or is a finite group, we can in fact use this to conclude that is contained in the Frattini subgroup of .