Editing P-group (section)

===Automorphisms===
The [[group automorphism|automorphism]] groups of ''p''-groups are well studied. Just as every finite ''p''-group has a non-trivial center so that the [[inner automorphism group]] is a proper quotient of the group, every finite ''p''-group has a non-trivial [[outer automorphism group]]<!-- Gaschuetz. In fact people tried to figure out how big it was, conjecture, minimal counterexample of order p^6, yada yada-->. Every automorphism of ''G'' induces an automorphism on ''G''/Φ(''G''), where Φ(''G'') is the [[Frattini subgroup]] of ''G''. The quotient G/Φ(''G'') is an [[elementary abelian group]] and its [[automorphism group]] is a [[general linear group]], so very well understood. The map from the automorphism group of ''G'' into this general linear group has been studied by [[William Burnside|Burnside]], who showed that the kernel of this map is a ''p''-group. <!-- give corresponding theorems for action on Ω and the socle -->