The first publication of Pierre Deligne was Congruences sur le nombre de sous-groupes d’ordre $p^k$ dans un groupe fini, Bull. Soc. Math. Belg. XVIII 2 (1966) pp. 129–132. I do not have access to this publication. What exactly did he prove here? Translated from the French the title means Congruences concerning the number of subgroups of order $p^k$ in a finite group.

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    $\begingroup$ You might want to consider accepting answers, which can be done by clicking the tick on the top left of answers. This gives the answerer an extra +15 rep. and shows that question is the one that served you most. Also, it shows the question is "answered", which helps users filter through open questions more easily. $\endgroup$ – Pedro Tamaroff Apr 19 '12 at 23:12
  • $\begingroup$ Done of course! $\endgroup$ – Nicky Hekster Apr 20 '12 at 8:01

I will paraphrase from MathSciNet review number MR0202821 (34 #2680), written by B. Chang:

Let $G$ be a group of order $p^sh$, with $(p,h)=1$ and let $d_k$ be the number of subgroups of $G$ of order $p^{s−k},\,\, 0\leq k\leq s.$ The main results are that:
1. $d_k=1 \mod p$
2. If a Sylow p-subgroup S is elementary abelian or cyclic, then $d_k$ is congruent mod $p^{k+1}$ to the number of subgroups of S of order $p^{s−k}$.
3. If S is not cyclic then $d_1=1+p \mod p^2$.

Note (July 26th 2018, NH) one can find the publication here!

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  • $\begingroup$ Jim, thanks very much!!! $\endgroup$ – Nicky Hekster Apr 19 '12 at 22:09

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