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?