Intersection of Sylow p-Subgroup with Subgroup not necessarily Sylow p-Subgroup
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Theorem
Let $G$ be a group.
Let $P$ be a Sylow $p$-subgroup of $G$.
Let $H$ be a subgroup of $G$.
Then $P \cap H$ is not necessarily a Sylow $p$-subgroup of $H$.
Proof
We note that from Intersection of Subgroups is Subgroup that $P \cap H$ is a subgroup of $G$ and also of $H$.
Let $G$ be the dihedral group $D_3$, given by its group presentation:
- $D_3 = \gen {a, b: a^3 = b^2 = e, a b = b a^{-1} }$
By definition of Sylow $p$-subgroup, $\gen a$ is a Sylow $3$-subgroup of $G$.
However, $\gen b$ is also a subgroup of $G$, of order $2$.
But:
- $\gen b \cap \gen a = e$
and $e$ is not a Sylow $3$-subgroup of $G$.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $11$: The Sylow Theorems: Exercise $3$