Centralizer and normalizer : Centralizer
In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively. These subgroups provide insight into the structure of G.
Definitions
The centralizer of an element a of a group G (written as CG(a)) is the set of elements of G which commute with a; in other words, CG(a) = {x in G : xa = ax}. If H is a subgroup of G, then CH(a) = CG(a) ∩ H. If there is no danger of ambiguity, we can write CG(a) as C(a).
More generally, let S be any subset of G (not necessarily a subgroup). Then the centralizer of S in G is defined as C(S) = (x in G : for all s in S, xs = sx}. If S = {a}, then C(S) = C(a).
C(S) is a subgroup of G; since if x, y are in C(S), then xy -1s = xsy -1 = sxy -1.
The center of a group G is CG(G), usually written as Z(G). The center of a group is both normal and abelian and has many other important properties as well. We can think of the centralizer of a as the largest (in the sense of inclusion) subgroup H of G having having a in its center, Z(H).
A releated concept is that of the normalizer of S in G, written as NG(S) or just N(S). The normalizer is defined as N(S) = {x in G : xS = Sx}. Again, N(S) can easily be seen to be a subgroup of G. The normalizer gets it name from the fact that if we let <S> be the subgroup generated by S, then N(S) is the largest subgroup of G having <S> as a normal subgroup (compare this with the conjugate closure of S).
Properties
If G is an abelian group, then the centralizer or normalizer of any subset of G is all of G; in particular, a group is abelian if and only if Z(G) = G.
If a and b are any elements of G, then a is in C(b) if and only if b is in C(a), which happens if and only if a and b commute. If S = {a} then N(S) = C(S) = C(a).
C(S) is always a normal subgroup of N(S): If c is in C(S) and n is in N(S), we have to show that n -1cn is in C(S). To that end, pick s in S and let t = nsn -1. Then t is in S, so therefore ct = tc. Then note that ns = tn; and n -1t = sn -1. So
- (n -1cn)s = (n -1c)tn = (n -1(tc)n = (sn -1)cn = s(n -1cn)
If H is a subgroup of G, then the N/C Theorem states that the factor group N(H)/C(H) is isomorphic to a subgroup of Aut(H), the automorphism group of H.
Since NG(G) = G, the N/C Theorem also implies that G/Z(G) is isomorphic to Inn(G), the subgroup of Aut(G) consisting of all inner automorphisms of G.
If we define a group homomorphism T : G → Inn(G) by T(x)(g) = Tx(g) = xgx -1, then we can describe N(S) and C(S) in terms of the group action of Inn(G) on G: the stabilizer of S in Inn(G) is T(N(S)), and the subgroup of Inn(G) fixing S is T(C(S)).
Then all will be straightened out."
"Walter," whispered Kennedy to me, "while we are waiting, you can
unfortunate man.html">man.html">man. When at last I had Thornton placed where no one
laboratory.
Craig was still there, waiting alone.
"That Dr. Lord will be a tough customer," he remarked. "Of course
caught the criminal. But that often is really only the beginning
however."
"I wish there was some elixir for fatigue," I remarked, as we
But instead of going to bed, Kennedy drew a book from the
deeply in his chair.
As for me, I went directly to my room, planning that to-morrow I
more important duties in order to appear before Dr. Leslie in the
for the Grand Jury, but it was not until nearly noon that Craig
sounded.
"A note.html">note for Mr. Kennedy," announced a man in a police uniform,
"harbour.html">Harbour Police, station.html">station.html">Station No. 3, Staten Island," was an urgent
out of the ordinary to interest you," I read when Kennedy tossed
to wait for us. "The Curtis family wish to retain a private
the death of Bertha Curtis, whose body was found this morning in
who had brought the note.
The Curtises, as we knew, were among the prominent families of
Curtis had been an actress, in spite of the means and social
result.
At the station of the harbour police, O'Connor and another man,
we had landed.
"There have been some queer doings about here," exclaimed the
introduce Mr. Walker Curtis."
In a lower tone as we walked up the dock O'Connor continued, "He
station found in the Kill this morning. They thought at first.
On
wordlookup.net
All is still licensed under the GNU FDL.
It uses material from the wikipedia.
|
|