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Center (group theory)
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<a href="/facts/Cayley_table/1L05nSE7">Cayley table</a> for <a href="/facts/Dihedral_group_of_order_8/JkT2jo1T">D4</a> showing elements of the center, {e, a2}, commute with all other elements (this can be seen by noticing that all occurrences of a given center element are arranged symmetrically about the center diagonal or by noticing that the row and column starting with a given center element are <a href="/facts/Transpose/8wmsagGS">transposes</a> of each other).<table><tbody><tr><th>∘</th><th>e</th><th>b</th><th>a</th><th>a2</th><th>a3</th><th>ab</th><th>a2b</th><th>a3b</th></tr><tr><th>e</th><td>e</td><td>b</td><td>a</td><td>a2</td><td>a3</td><td>ab</td><td>a2b</td><td>a3b</td></tr><tr><th>b</th><td>b</td><td>e</td><td>a3b</td><td>a2b</td><td>ab</td><td>a3</td><td>a2</td><td>a</td></tr><tr><th>a</th><td>a</td><td>ab</td><td>a2</td><td>a3</td><td>e</td><td>a2b</td><td>a3b</td><td>b</td></tr><tr><th>a2</th><td>a2</td><td>a2b</td><td>a3</td><td>e</td><td>a</td><td>a3b</td><td>b</td><td>ab</td></tr><tr><th>a3</th><td>a3</td><td>a3b</td><td>e</td><td>a</td><td>a2</td><td>b</td><td>ab</td><td>a2b</td></tr><tr><th>ab</th><td>ab</td><td>a</td><td>b</td><td>a3b</td><td>a2b</td><td>e</td><td>a3</td><td>a2</td></tr><tr><th>a2b</th><td>a2b</td><td>a2</td><td>ab</td><td>b</td><td>a3b</td><td>a</td><td>e</td><td>a3</td></tr><tr><th>a3b</th><td>a3b</td><td>a3</td><td>a2b</td><td>ab</td><td>b</td><td>a2</td><td>a</td><td>e</td></tr></tbody></table> <p>In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, the center of a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> <i>G</i> is the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of elements that <a href="/facts/Commutative/WaYxJLxd">commute</a> with every element of <i>G</i>. It is denoted Z(<i>G</i>), from German <i>Zentrum,</i> meaning <i>center</i>. In <a href="/facts/Set-builder_notation/0nVHQF6E">set-builder notation</a>, </p> Z(<i>G</i>) = {<i>z</i> ∈ <i>G</i> | ∀<i>g</i> ∈ <i>G</i>, <i>zg</i> = <i>gz</i>}. <p>The center is a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a>, Z(<i>G</i>) ⊲ <i>G</i>, and also a <a href="/facts/Characteristic_subgroup/UfoS7U7S">characteristic</a> subgroup, but is not necessarily <a href="/facts/Fully_characteristic_subgroup/UfoS7U7S">fully characteristic</a>. The <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a>, <i>G</i> / Z(<i>G</i>), is <a href="/facts/Group_isomorphism/LC4N1aDw">isomorphic</a> to the <a href="/facts/Inner_automorphism/awJPzhDa">inner automorphism</a> group, Inn(<i>G</i>). </p><p>A group <i>G</i> is abelian if and only if Z(<i>G</i>) = <i>G</i>. At the other extreme, a group is said to be centerless if Z(<i>G</i>) is <a href="/facts/Trivial_group/XPjQkQFm">trivial</a>; i.e., consists only of the <a href="/facts/Identity_element/9nss3xHY">identity element</a>. </p><p>The elements of the center are central elements. </p>