Center (group theory)

The center of G is always a subgroup of . In particular: • contains the identity element of , because it commutes with every element of , by definition: , where is the identity; • If and are in , then so is , by associativity: for each ; i.e., is closed; • If is in , then so is as, for all in , commutes with : . Furthermore, the center of is always an abelian and normal subgroup of . Since all elements of commute, it is closed under conjugation. A group homomorphism might not restrict to a homomorphism between their centers. The image elements commute with the image , but they need not commute with all of unless is surjective. Thus the center mapping G\to Z(G) is not a functor between categories Grp and Ab, since it does not induce a map of arrows. ==Conjugacy classes and centralizers==

By definition, an element is central whenever its conjugacy class contains only the element itself; i.e. {{math|1=Cl(g) = {g}}}. The center is the intersection of all the centralizers of elements of : Z(G) = \bigcap_{g\in G} Z_G(g). As centralizers are subgroups, this again shows that the center is a subgroup. == Conjugation ==

Consider the map , from to the automorphism group of defined by , where is the automorphism of defined by :. The function, is a group homomorphism, and its kernel is precisely the center of , and its image is called the inner automorphism group of , denoted . By the first isomorphism theorem we get, :. The cokernel of this map is the group of outer automorphisms, and these form the exact sequence :. ==Examples==

• The center of an abelian group, , is all of . • The center of the Heisenberg group, , is the set of matrices of the form: \begin{pmatrix} 1 & 0 & z\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} • The center of a nonabelian simple group is trivial. • The center of the dihedral group, , is trivial for odd . For even , the center consists of the identity element together with the 180° rotation of the polygon. • The center of the quaternion group, {{math|1=Q = {1, −1, i, −i, j, −j, k, −k} }}, is {{math|{1, −1}}}. • The center of the symmetric group, , is trivial for . • The center of the alternating group, , is trivial for . • The center of the general linear group over a field , , is the collection of scalar matrices, {{math|{{mset| sIn ∣ s ∈ F \ {0} }}}}. • The center of the orthogonal group, is {{math|{In, −In}}}. • The center of the special orthogonal group, is the whole group when , and otherwise when n is even, and trivial when n is odd. • The center of the unitary group, U(n) is \left\{ e^{i\theta} \cdot I_n \mid \theta \in [0, 2\pi) \right\}. • The center of the special unitary group, \operatorname{SU}(n) is \left\lbrace e^{i\theta} \cdot I_n \mid \theta = \frac{2k\pi}{n}, k = 0, 1, \dots, n-1 \right\rbrace . • The center of the multiplicative group of non-zero quaternions is the multiplicative group of non-zero real numbers. • Using the class equation, one can prove that the center of any non-trivial finite p-group is non-trivial. • If the quotient group is cyclic, is abelian (and hence , so is trivial). • The center of the Rubik's Cube group consists of two elements – the identity (i.e. the solved state) and the superflip. The center of the Pocket Cube group is trivial. • The center of the Megaminx group has order 2, and the center of the Kilominx group is trivial. ==Higher centers==

Quotienting out by the center of a group yields a sequence of groups called the upper central series: : The kernel of the map is the th center of (second center, third center, etc.), denoted . Concretely, the ()-st center comprises the elements that commute with all elements up to an element of the th center. Following this definition, one can define the 0th center of a group to be the identity subgroup. This can be continued to transfinite ordinals by transfinite induction; the union of all the higher centers is called the hypercenter. The ascending chain of subgroups : stabilizes at i (equivalently, ) if and only if is centerless. Examples • For a centerless group, all higher centers are zero, which is the case of stabilization. • By Grün's lemma, the quotient of a perfect group by its center is centerless, hence all higher centers equal the center. This is a case of stabilization at . ==See also==