Automorphisms of FInite Groups

The leitmotif for this book is the observation that “the symmetries of a group G are encoded in the automorphism group \( Aut(G) \) of \( G\),” the focus falling on finite groups \( G \).  The authors consider a number of interesting questions surrounding this theme, including what they refer to as the Ledermann-Neumann theorem and the Divisibility Problem.  We find out from the book’s Preface that the result of Ledermann and Neumann, dating back to 1956, constructs a cubic polynomial \( f \) with coefficients in the natural numbers with the beautiful property that if \( p \) is any prime, \( h \) any natural number, and \( G \) is a finite group whose order is divisible by \( p^{f(h)} \) then \( p^{h} \) divides the order of \( G \)’s automorphism group.  The way to look at this result is to ask the question, given the requirement that the order of  \( Aut(G) \) should be divisible by \( p^{h} \), what can be said about the order of \( G \)?  How \( p \)-small can G be so as to force, still, that the automorphism group of \( G \) should have order divisible by \( p^{h} \)?  In other words, we’re looking for lower bounds on number theoretic functions \( f \) such that  \( p^{f(h} | o(G) \) implies \( p^{h} |o(Aut\;G) \).  Here there is a result in place to the effect that we need \( f(h) \geq h-1 \) (established by Hyde in 1970).  Say the authors: “One might wonder whether this bound can be lowered further if we restrict ourselves to … finite p-groups.”

  

Together with the foregoing considerations consider the aforementioned Divisibility Problem, phrased as follows: given any non-cyclic finite \( p \)-group, \( G \), does the identity function \( i:= i|_{N\backslash \{1,2\} } \) satisfy the desired property, namely, that if \( p^{i(h)} | o(G) \) then \( p^{h}|o(Aut\;G) \)?  Well, this is where things do get a little sticky: in 2015 it was shown by Gonzáles-Sanchez and Jaikin-Zapirain that there actually exist non-cyclic finite \( p \)-groups of order \( > p^{2} \) with the property that this order fails to divide the order of the corresponding automorphism groups: counterexamples to the foregoing claim.   “We [the authors of the book under review] present a detailed exposition of this important development in the theory of automorphism groups.  However, it is still intriguing to know for what other classes of non-cyclic finite p-groups the problem has an affirmative solution and to construct explicit counterexamples.”  In order to further this cause the authors stipulate that “[a] non-cyclic finite \( p \)-group \( G \) of order \( > p^{2} \) is said to have the Divisibility Property if [ \( o(G)| o(Aut\;G) \)].”  They add, not surprisingly, that “determining all  finite \( p \)-groups admitting [the] Divisibility Property continues to be a challenging problem.”  

 

The authors present a few other themes of a very similar flavor: the tenor of what they are up to is amply illustrated, however, by the preceding examples.

 

The discussion of all these themes is split into three parts, respectively concerning Wells’ exact sequence, the number-theoretic function \( f \) discussed above, and the foregoing Divisibility Property.  It is proper, indeed, to start with a thorough discussion of the Wells exact sequence, seeing that this result in the cohomology of groups is concerned with the question of extending automorphisms of the kernel or image groups in a short exact sequence to the central term: clearly a very useful tool in the kind of analysis the authors propose. Thus, the Wells exact sequence is attached to any given s.e.s. of finite groups.  Obviously, the question of extending and lifting automorphisms in the setting of short exact sequences is, in itself, of independent interest.

 

With these three parts delineated, the book’s first two chapters introduce \( p \)-groups and the Wells exact sequence, the next chapter deals with the Ledermann-Neumann theorem, and the final three chapters go at the Divisibility Property.  The audience for this interesting book includes group theorists and graduate students headed in this direction.  The authors propose that prerequisites for reading this work include “an advanced course in the theory of finite groups,” as well as some more exotic stuff needed in particular places in their book: for example, the book’s last chapter requires “basic analysis, topology, and Lie algebras.” 

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Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.