- Center (group theory)
In
abstract algebra , the center of a group "G" is the set "Z"("G") of all elements in "G" which commute with all the elements of "G". That is,:.
Note that "Z"("G") is a
subgroup of "G", because
# "Z"("G") contains "e", theidentity element of "G", because "eg" = "g" = "ge" for all "g" ∈ G by definition of "e", so by definition of "Z"("G"), "e" ∈ "Z"("G");
# If "x" and "y" are in "Z"("G"), then ("xy")"g" = "x"("yg") = "x"("gy") = ("xg")"y" = ("gx")"y" = "g"("xy") for each "g" ∈ "G", and so "xy" is in "Z"("G") as well (i.e., "Z"("G") exhibits closure);
# If "x" is in "Z"("G"), then "gx" = "xg", and multiplying twice, once on the left and once on the right, by "x"−1, gives "x"−1"g" = "gx"−1 — so "x"−1 ∈ "Z"("G").Moreover, "Z"("G") is an abelian subgroup of "G", a
normal subgroup of "G", and even a strictlycharacteristic subgroup of "G", but not always fully characteristic.The center of "G" is all of "G"
if and only if "G" is an abelian group. At the other extreme, a group is said to be centerless if "Z"("G") is trivial, i.e. consists only of the identity element.Conjugation
Consider the map "f": "G" → Aut("G") from "G" to the
automorphism group of "G" defined by "f"("g") = "g", where "g" is the automorphism of "G" defined by :.This is agroup homomorphism , and its kernel is precisely the center of "G", and its image is called theinner automorphism group of "G", denoted Inn("G"). By thefirst isomorphism theorem we get:Thecokernel of this map is the group ofouter automorphism s, and these form theexact sequence ::Examples
* The center of the group of "n"-by-"n" invertible matrices over the field is the collection of scalar matrices .
* The center of theorthogonal group is .
* The center of thequaternion group is .
* The center of the multiplicative group of non-zeroquaternion s is the multiplicative group of non-zero real numbers.
* Using theclass equation one can prove that the center of any non-trivial finitep-group is non-trivial.
* Non-abeliansimple group s are centerless.
* If thequotient group is cyclic, G is abelian.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 "i"th center of "G" (second center, third center, etc.), and is denoted .Following this definition, one can define the 0th center of a group to be the identity subgroup. This can be continued totransfinite ordinals bytransfinite induction ; the union of all the higher centers is called thehypercenter . [This union will include transfinite terms if the UCS does not stabilize at a finite stage.]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.
* ByGrün's lemma , the quotient of aperfect group by its center is centerless, hence all higher centers equal the center. This is a case of stabilization at .References
ee also
*
center (algebra)
*centralizer and normalizer
*conjugacy class .
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