Many theorems about odd order $p$-groups fail miserably for $2$-groups. These can range from simple $2$-group exceptions (e.g. Frobenius complements can be either cyclic or generalized quaternion) to full blown analogs proved with vastly different, "$2$-groupy" techniques (e.g. Glauberman's $\text{ZJ}$ theorem vs. Stellmacher's results about $\Sigma_4$-free groups.) It's clear that $2$-groups in some way work fundamentally differently than other $p$-groups, and I would like to improve my understanding of exactly how.

Does anyone know of a comprehensive reference compiling important results about $2$-groups specifically? Is there a book or survey article about the theory of $2$-groups out there somewhere?

I would be especially interested in sources discussing differences in the internal structure of $2$-groups, rather than differences associated with their place in finite groups, such as my examples above. (And again, I don't need any references for $p$-groups in general - I've got plenty of those.)

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    $\begingroup$ Berkovich and Janko, while ostensibly about $p$-groups in general, devote more than half their three books to results on $2$-groups. $\endgroup$
    – user641
    Mar 1, 2013 at 7:42
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    $\begingroup$ This is a good question. I too have many times been amazed at how different $p$-groups behave when $p=2$ (a good example where the complications show up en masse is the classification of $p$-groups all of whose non-normal subgroups are cyclic, which is chapter 16 in Berkovich). $\endgroup$ Mar 1, 2013 at 14:22
  • $\begingroup$ You may search about Suzuki 2-groups. There are interesting results on them. $\endgroup$
    – S.B.
    Jul 28, 2015 at 13:43

1 Answer 1


There is a book on the theory of $p$-groups that covers $2$-groups well. Check http://web.mat.bham.ac.uk/D.A.Craven/docs/lectures/pgroups.pdf.


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