Frattini subgroup

In group theory, the Frattinigruppe (or more accurately Frattiniuntergruppe ) is a special subgroup of a given group. With their help can be examined in particular the structure of finite p-groups. It is named after the Italian mathematician Giovanni Frattini.

Definition

Is a group, then the Frattinigruppe is defined as the average of all maximal subgroups.

This is called a subgroup of maximal if true and there are no real major subgroup.

If no maximal subgroups has, as in the case of the trivial group or some infinite groups such as the audit team, is one.

Properties

• The Frattinigruppe is a characteristic subgroup, ie in particular a normal subgroup.
• Is finite, then is nilpotent. Is also nilpotent, then is nilpotent.
• Applies with a subset of, then.
• The Frattinigruppe consists exactly of the non- producers of, that is, it holds if and only if for every subset follows from always. In other words, the members of Frattinigruppe are in any system of generators superfluous.