Identity of Group is in Center

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Theorem

Let $G$ be a group.

Let $e$ be the identity of $G$.


Then $e$ is in the center of $G$:

$e \in \map Z G$


Proof

From Center is Intersection of Centralizers:

$\displaystyle \map Z G = \bigcap_{g \mathop \in G} \map {C_G} g$

where $\map {C_G} g$ denotes the centralizer of $g$.

From Centralizer of Group Element is Subgroup, each of $\map {C_G} g$ is a subgroup of $G$.

From Identity of Subgroup:

$\forall g \in G: e \in \map {C_G} g$

Hence by definition of set intersection:

$e \in \displaystyle \bigcap_{g \mathop \in G} \map {C_G} g$

whence the result.

$\blacksquare$