Pushing-up amalgam

Origin of the concept
The concept of pushing-up amalgam was introduced in the context of questions like: under what conditions do we have a Sylow subgroup for which no nontrivial subgroup is normal in the whole group?

Origin of the term
The term has been introduced by George Glauberman in his work on the classification of finite simple groups.

Definition with symbols
This definition has been taken from the paper A Note on Pushing up by Paul Fan.

Suppose $$P$$ and $$H$$ are abstract finite groups. A pushing-up amalgam for them is a group $$S$$ with embeddings in both $$P$$ and $$H$$, such that:


 * $$S$$ is a Sylow subgroup of $$P$$.
 * $$S$$ is normal in $$H$$.
 * No nontrivial subgroup of $$S$$ is normal in both $$P$$ and $$H$$.