p-Subgroup

Beschreibung

A $p$-Subgroup is a Subgroup, whose order is a power of a primme number $p$. It is a generalization of a p-Sylowgruppe.

Eigenschaften

Theorem (Relationship between context group and normalizer):

If $H$ is a $p$-subgroup of $G$ then $[N_G(H):H]{\equiv }_p[G:H]$

An equivalent way of stating this is that the number of cosets of $H$ outside of $N_G(H)$ is a multiple of $p$.

Proof: Let $S$ be the set of cosets of $H\subseteq G$. We will study the action of $H$ on $S$ by leftmultiplication $h\cdot gH=hgH$. This seem like it would always fix the coset but this is actuallu only the case if $H$ is a Normalteiler. Meaning, the cosets $gH$ fixed by the action are exactly the cosets in the Normalisator (Gruppe) $N_G(H)$. The number of those cosets if given by $[N_G(H):H]$. A theorem in the Class equation tells us, that the number of fixed elements is congruent mod $p$ to the size of $S$. $\square$

Beispiele

Beispiel: