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In a recent preprint 1302.1929 my co-authors and I make use of some results about the character theory of extraspecial groups. These were largely gleaned from haphazard Googling, comments made by people on MO, and playing around by ourselves.

Because the paper is probably going to be read by people in Banach algebras/harmonic analysis rather than by finite group theorists, the referee has requested that we provide references for the following assertions: if $G$ is a group with prime power order where the centre and derived subgroup coincide and have order $p$, for $p$ a prime, then each non-linear irreducible character has degree $p^n$ where $|G|=p^{2n+1}$, and is supported on $Z(G)$.

Now while this does not look hard to prove directly, just using fairly basic character theory, I would like to know if this is written down explicitly somewhere where I can cite it. The Wikipedia page for extraspecial groups lists Gorenstein's book on finite groups as a reference, but my library doesn't have a copy.

Can anyone suggest a suitable reference? (I would like to check that no such convenient reference exists before writing out a proof in the paper.)

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  • $\begingroup$ And before anyone suggests it, even in jest, I don't think they will let me cite Wikipedia... $\endgroup$ – Yemon Choi Jun 4 '13 at 18:12
  • $\begingroup$ Both answers are good for my purposes, but since I find Isaacs's argument (which proves less and is therefore shorter) easier to follow, I've accepted Carlo Beenakker's suggestion $\endgroup$ – Yemon Choi Jun 7 '13 at 16:34
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I'm sure you'll correct me quickly if I'm off the mark, but isn't this proven in the appendix of Threads Through Group Theory? (article by Persi Diaconis, appendix by Marty Isaacs, published in "Character Theory of Finite Groups".)

Quite apart from the proof itself, the context how it ended up in this appendix is quite interesting, as Diaconis writes in the intro:

Marty Isaacs believes in answering questions. They can come from students, colleagues, or perfect strangers. As long as they seem serious, he usually gives it a try. This paper records the path of a letter that Marty wrote to a stranger (me). It led to more correspondence and a growing set of extensions. All of this work can be traced back to Marty’s letter. [...] We are still meeting, writing, and following the thread together. I am truly thankful.

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  • $\begingroup$ Hmm, it looks good on a quick skim. In fact, I have an embarrassing suspicion that I may have read this very bit by Isaacs while Google-grazing two years ago... $\endgroup$ – Yemon Choi Jun 4 '13 at 19:15
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Theorem 5.5 of Chapter 5 of Gorenstein's book:

Gorenstein, D., Finite Groups, Harper and Row, New York, 1968.

states:

Let $P$ be an extra-special $p$-group of order $p^{2r+1}$ and let $F$ be a field of characteristic $0$ or prime to $p$ which contains a primitive $p^2$-root of unity. Then the faithful irreducible representations of $P$ over $F$ are all of degree $p^r$.

That seems to be exactly what you want, and you should be able to find it in http://books.google.co.uk/.

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  • $\begingroup$ I'm probably being dense, but it isn't clear to me why this result precludes the existence of (non-faithful) non-linear irreps of some other degree $\endgroup$ – Yemon Choi Jun 4 '13 at 19:16
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    $\begingroup$ The non-faithful ones must have $Z(G)$ in their kernel (because the kernel must intersect $Z(G)$ non-trivially and $|Z(G)|=p$), and since $Z(G) = G'$, they are representations of the abelian group $G/Z(G)$, and hence they are linear. $\endgroup$ – Derek Holt Jun 4 '13 at 19:42

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