= Center of a group
{title2=$Z(G)$}
{wiki=Center_(group_theory)}
= Centre of a group
{synonym}
= Group center
{synonym}
= Group centre
{synonym}
The center of a <group> consists of the elements that commute with every element of that <group>:
$$
Z(G)=\{z\in G:zg=gz\text{ for every }g\in G\}=\bigcap_{g\in G}C_G(g),
$$
where $C_G(g)$ is the <centralizer> of $g$. These are operations whose position in any product can be changed freely. The center is an <abelian group> and a <characteristic subgroup>, hence a <normal subgroup>. A <group> is an <abelian group> precisely when $Z(G)=G$.
For the <symmetric group> $S_3$, the <conjugacy classes> are
$$
\{e\},\qquad\{(12),(13),(23)\},\qquad\{(123),(132)\}.
$$
An element belongs to the center precisely when its <conjugacy class> has one element, so $Z(S_3)=\{e\}$. Yet <commutator subgroup of the symmetric group on three letters> gives $S_3'=A_3$, with three elements. Thus belonging to the <commutator subgroup> does not mean commuting with everything. The <alternating group> $A_3$ is an <abelian group> as a <subgroup>, but its nonidentity elements do not commute with the <transpositions> in the surrounding $S_3$.
For the <dihedral group> $D_8$ of <group order> eight in <abelianization of the square symmetry group>, the center is $\{e,r^2\}$. A <rotation> $r^i$ commutes with $s$ precisely when $r^i=r^{-i}$, which requires $i$ even. No <reflection> $r^is$ commutes with $r$, since $(r^is)r=r^{i-1}s$ whereas $r(r^is)=r^{i+1}s$. Consequently
$$
Z(D_8)=\{e,r^2\}=D_8'.
$$
The half-turn can be performed before or after any <square> symmetry with the same result. Its appearance both here and in the <commutator subgroup> is a feature of this example, rather than a general equality. See https://kconrad.math.uconn.edu/blurbs/grouptheory/dihedral.pdf[Conrad, Dihedral groups I, theorem 3.4 and example 3.5].
The center and the <commutator subgroup> answer different useful questions: which operations already commute with everything, and which operations must disappear for the entire <group> to commute. In an <abelian group> these answers are $Z(G)=G$ and $G'=\{e\}$. Another way to use the center is through <conjugation>: the <group homomorphism> $g\mapsto(h\mapsto ghg^{-1})$ has <group kernel> $Z(G)$, so the <first isomorphism theorem> gives $G/Z(G)\cong\operatorname{Inn}(G)$, the <inner automorphism group>. The <quotient group> by the center thus records the distinct conjugation operations; it need not be an <abelian quotient>.
A nontrivial finite <p-group> always has a nontrivial center. In the <class equation>, each noncentral <conjugacy class> has size divisible by $p$, so $|Z(G)|\equiv|G|\equiv0\pmod p$. The <identity element> ensures $Z(G)$ is nonempty, hence it has at least $p$ elements. See https://ocw.mit.edu/courses/18-703-modern-algebra-spring-2013/96221ce2d5b1077b3e5d3c5e08c88afe_MIT18_703S13_pra_l_23.pdf[McKernan, MIT 18.703, lemmas 23.6–23.7 and definition-lemma 23.10].
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