# 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?

• 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. Jul 5, 2010 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? Jul 6, 2010 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. Jul 21, 2010 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. Aug 9, 2012 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.