---- > [!definition] Definition. ([[center of a group]]) > The center of a [[group]] $G$, denoted $Z(G)$ is the set of elements $z$ in $G$ that commute with every element of $G$. It is a [[subgroup]] of $G$. > [!justification] > We need to show that $Z(G) \leq G$. $Z(G)=G$ for $|G|<3$ ([[classification of small groups|groups of order less than 6 are abelian]]), so assume $|G|\geq 3$. Let $e$ denote the identity element of $G$. Since $ea=a=ae$ for all $a \in G$, we have $e \in Z(G)$. Next, suppose $b,c \in Z(G)$, so that for all $a \in G$ we have $ab=ba$ and $ac=ca$. Then the computation $\begin{align} a(bc)= & (ab)c \\ = & (ba)c \\ = & b(ac) \\ = & b(ca) \\ = & (bc)a \end{align}$ shows $Z(G)$ is closed under $Gs [[binary operation|group operation]]. Finally, we must show closure of $Z(G)$ under taking inverses. We have $\begin{align} b ^{-1}a= & b^{-1}a(bb ^{-1})\\= & b^{-1}(ab)b^{-1}\\= & b ^{-1}(ba)b^{-1} \\= & (b^{-1}b)(ab^{-1})=ab^{-1}, \end{align}$ as required. > [!basicexample] Example. ($Z(D_{n})$) > Let $D_{n}$ denote the [[dihedral group]] of rigid symmetries of the regular $n$-gon. Claim:$Z(D_{n})=\begin{cases} \{ e \} & n \text{ odd}, \\ \{ e, x^{n/2} \} & n \text{ even}. \end{cases}$ \ **Proof of Claim.** By definition, $e$ commutes with all elements of $D_{n}$, so we will focus on the case where $n$ is even, and demonstrate in this case that $x^{n/2}$ commutes with each elements of $D_{n}$. \ So let $a \in D_{n}$ be arbitrary. We know $a=y^{i}x^{j}$ for some $i \in [2]$ and $j\in \{ 0 \} \cup [n-1]$. \ Suppose $i=2$. Then $a=x^{j}$ and we have $ax^{n/2}=x^{j}x^{n/2}=x^{j+n/2}=x^{n/2+j}=x^{n/2}x^{j}=x^{n/2}a$as required. \ Suppose $i=1$; then $a=yx^{j}$. First note that, using the given [[dihedral group|dihedral group relation]] and that $y=y^{-1}$, we have $\begin{align} xy= & (yy^{-1})(xy) \\ = & y(y^{-1} xy) \\ = & y(yxy^{-1}) \\ = & yx ^{-1}. \end{align}$ Using this and the fact that $x^{n/2}=x^{-n/2}$, we can compute $\begin{align} x ^{n/2}a & = x^{n/2}yx^{j} \\= &\overbrace{x \cdots x}^{n/2 \text{ times }}y x ^{j} \\ = & \overbrace{x \cdots x}^{(n-1)/2 \text{ times }}\overbrace{xy}^{yx ^{-1}} x^{j} \\ = & \overbrace{x \cdots x}^{(n-2)/2 \text{ times }} \overbrace{xy \ x ^{-1}}^{yx^{-2}} x ^{j}\\ = \\ \vdots \\ = & yx^{-n/2} x^{j} \\ = & yx^{n/2} x ^{j} \\ = & a x ^{n/2} \end{align}.$ as desired. \ \ The geometric interpretation of $Z(D_{n})$ as follows. In general: $\text{reflection followed by $k$-\textcolor{Skyblue}{\textcolor{Skyblue}{counter-clockwise}} rotation } $$ \text{ behaves like $k$-\textcolor{Thistle}{clockwise} rotation followed by reflection.}$ Since there is no difference between rotating counter-clockwise by $180^{\circ}$ and rotating clockwise by $180^{\circ}$, for the case of $x^{n/2}$ we can simplify the above statement into $\text{reflection followed by \textcolor{Skyblue}{\textcolor{Skyblue}{{$\cancel{\text{counter-clockwise}}$}}} rotation ~~} $$ \text{ behaves like \textcolor{Thistle}{$\cancel{\text{clockwise}}$} rotation followed by reflection,}$ i.e., we get that reflection$\to$rotation is identical to rotation$\to$reflection. Since all group elements are compositions of rotations and reflections, the commutativity for this special $x^{n/2}$ is clear. \ Of course, there are no $180^{\circ}$ rotations when $n$ is odd, and so there is correspondingly no non-trivial center for $G$ when $n$ is odd. ^accce0 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```