Center of Opposite Group
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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.
$\blacksquare$