Center of Opposite Group

Theorem

Let $\struct {G, \circ}$ be a group.

Let $\struct {G, *}$ be the opposite group to $G$.

Let $\map Z {G, \circ}$ and $\map Z {G, *}$ be the centers of $\struct {G, \circ}$ and $\struct {G, *}$, respectively.

Then:

$\map Z {G, \circ} = \map Z {G, *}$

Proof

We have, for $g \in G$:

\(\ds g \in \map Z {G, \circ}\)

\(\leadstoandfrom\)

\(\ds \forall h \in G: g \circ h = h \circ g\)

Definition of Center of Group

\(\ds \)

\(\leadstoandfrom\)

\(\ds \forall h \in G: h * g = g * h\)

Definition of Opposite Group

\(\ds \)

\(\leadstoandfrom\)

\(\ds g \in \map Z {G, *}\)

Definition of Center of Group

Hence the result, by definition of set equality.

$\blacksquare$