# Why are p-elementary groups so crucial in finite group theory?

I see what makes finite p-groups such a nice thing to study: non-trivial center, lots of interesting decreasing and increasing series coming from the p-power map etc.

To me it seems natural to expect huge difficulties when generalizing a result from p-groups to all finite groups. There surely are lots of ways to "go up a little bit" in the direction of non-p-ness. The direct product with a cyclic-group of order prime to p is the smallest step on this ladder.

Here comes the question: Why are often (representation theoretic) results for general finite groups a direct consequence of the special case for p-elementary groups?

Or am I mistaken with this impression?

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The only thing I know about $p$-elementary groups is that every (complex?) representation of every finite group can be made by adding and subtracting representations induced from such groups. Pretty clearly this is a strong tool for reducing statements to the case of $p$-elementary groups. – Tom Goodwillie Jul 5 '10 at 22:16
Note that a $p$-elementary group is a direct product of a $p$-group and a cyclic group of order prime to $p$. As Tom indicates, Brauer's theorem says that any complex character of a finite group is a $\mathbb{Z}$-linear combination of characters induced from $p$-elementary subgroups for various $p$ (found in many books including Chap. 10 of Serre). But Konrad's assertion about results for arbitrary groups often being a direct consequence of results for $p$-elementary groups is way overstated. Are there examples of such? – Jim Humphreys Jul 6 '10 at 16:04
Just a short warning for those not so familiar with the area (myself included). A p-elementary group is the product of a p-group P and a cyclic group of order prime to p. This is different from an elementary abelian p-group which is a product of cyclic groups of order p. Elementary abelian p-groups also play an important role, for instance when you study fusion systems. But that's a different story. – James Griffin Jul 21 '10 at 16:13
@Jim Humphreys: This 'often .. direct' surely was over the top. At that time I was thinking about Kakde's results on K_1 of group rings using the following reduction steps: p-adic Lie groups -> finite groups -> l-hyperelementary group, for all l -> p-hyperelementary groups -> p-elementary -> p-groups. I was under the impression, that his (or Oliver's in 'Whitehead grps of fin grps') were relying only on general facts for G-modulations. Since I have no particular knowledge in finite group theory, to conclude a general principle from this was optimistic, for sure. – Konrad Aug 9 '12 at 13:37

Perhaps this is not a direct answer to your question, but one of my favorite examples of $p$-groups entering into the theory of general finite groups (except of course for Sylow theory itself!) occurs in a theorem due to Frobenius. This theorem states:

Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Then the following are equivalent:

(i) $G$ has a normal $p$-complement.

(ii) $N_G(U)$ has a normal $p$-complement for all $p$-subgroups $U\subseteq G$ with $U>1$.

(iii) $N_G(U)/C_G(U)$ is a $p$-group for all $p$-subgroups $U\subseteq G$.

(iv) There is no fusion in $P$.

(Here $C_G(U)$ and $N_G(U)$ denote the centralizer and normalizer of the subgroup $U\subseteq G$, respectively. The quotient $N_G(U)/C_G(U)$ makes sense, of course, since $C_G(U)$ is normal in $N_G(U)$. Also, to say that there is no fusion in $P$ is to say that any pair of elements in $P$ that are $G$-conjugate are already $P$-conjugate.)

A remarkable theorem of Thompson asserts that, if $p\neq 2$ in Frobenius' theorem, to verify that $G$ has a normal $p$-complement is tantamount to checking that $N_G(U)$ has a normal $p$-complement for just two particular $p$-subgroups $U\subseteq G$. The two subgroups $U\subseteq G$ whose normalizers need to be checked are $Z(P)$ and $J(P)$ where $P$ is a fixed Sylow $p$-subgroup of $G$ and $Z(P)$ and $J(P)$ are the center and Thompson subgroup of $P$, respectively.

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