@ARTICLE{GROUP_22.MIZ,
  AUTHOR = {Nelson, Alexander M.},
  TITLE = {Characteristic Subgroups},
  ACKNOWLEDGEMENT = {The author would like to thank Adam Grabowski for
  his invaluable help and kindness.},
  SECTION1 = {Preparatory Work},
  SECTION2 = {Nontrivial Groups and Subgroups},
  SECTION3 = {Proper Subgroups},
  SECTION4 = {Automorphisms},
  SECTION5 = {Inner Automorphisms},
  SECTION6 = {Characteristic Subgroups},
  SECTION7 = {Appendix 1: Results concerning meets},
  SECTION8 = {Appendix 2: Centralizer of Characteristic Subgroups is Characteristic},
  DAY = {23},
  MONTH = {July},
  YEAR = {2022},
  ADDRESS1 = {Los Angeles, California, United States of America},
  SUMMARY = {We formalize the notion of characteristic subgroups using
  the definition found in Dummit and Foote~\cite{dummit-foote:2004}, as
  subgroups invariant under automorphisms from its parent group. Along
  the way, we formalize notions of Automorphism and results concerning
  centralizers. Much of what we formalize may be found sprinkled throughout  
  the literature, in particular Gorenstein~\cite{gorenstein:1980}
  and Isaacs~\cite{isaacs:2008}. We show all our favorite subgroups turn
  out to be characteristic: the center, the derived subgroup, the
  commutator subgroup generated by characteristic subgroups, and the
  intersection of all subgroups satisfying a generic group property.}}

@book{dummit-foote:2004,
  title={{Abstract Algebra}},
  author={Dummit, David S and Foote, Richard M},
  edition={{Third}},
  year={2004},
  publisher={Wiley and Sons}
}
@book{gorenstein:1980,
  title={{Finite Groups}},
  author={Gorenstein, Daniel},
  year={1980},
  edition={{Second}},
  publisher={Chelsea Publishing Company}
}
@book{isaacs:2008,
  author={Isaacs, I.~Martin},
  title={{Finite Group Theory}},
  year={2008},
  publisher={American Mathematical Society},
  series={Graduate Studies in Mathematics},
  volume={92}
}
