Centralizer in Subgroup is Intersection
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Theorem
Let $G$ be a group.
Let $H$ be a subgroup of $G$.
Then:
- $\forall x \in G: \map {C_H} x = \map {C_G} x \cap H$
That is, the centralizer of an element in a subgroup is the intersection of that subgroup with the centralizer of the element in the group.
Proof
It is clear that:
- $g \in \map {C_H} x \iff g \in \map {C_G} x \land g \in H$
The result follows by definition of set intersection.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Proposition $10.19$
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Proposition $10.25$