Talk:Centralizer and normalizer/Archive 1

The statement:

• The normalizer gets its 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.

is incorrect.

Let H = < s | s³ = 1 > the cyclic group of order 3.

Let G = <s, t | s³ = 1 , t^-1st =s² > an HNN extention of H which embedds H in the obvious way.

Let S = {s}. Then t^-1st is not in S so t is not in N_S(G). However it is contained in N_H(G), which (since H=<S>) is the largest subgroup of G having <S> as a normal subgroup. Bernard Hurley 21:50, 6 October 2006 (UTC)

Typos

I don't want to make the edit myself, in case I am mistaken, but in the first sentance:

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.

Shouldn't it infact read:

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 G as a whole, respectively. These subgroups provide insight into the structure of G.

James.robinson (talk) 07:46, 1 January 2009 (UTC)

No, the first one is correct. The "on the elements of S" refers to the centraliser, and "S as a whole" is the normaliser - check the defs for clarification SetaLyas (talk) 22:48, 18 March 2009 (UTC)

Lie algebras

There are analogous, but nonidentical, notions of centralizer and normalizer in Lie algebras. 99.231.65.91 (talk) 21:23, 24 January 2009 (UTC)

disambig "centralizer"

A centralizer is also a tool, e.g. in oil drilling.[1] —Preceding unsigned comment added by 92.78.99.50 (talk) 21:24, 27 November 2010 (UTC)

Reference/sources

There is a reference at the end of the article, so it seems to me that the frightener at the beginning declaring that there are none needs to be removed. --Brian Josephson (talk) 21:25, 22 December 2011 (UTC)

Normalizer is NOT always a subgroup of G

See http://www.markrobrien.com/hw3sol.pdf - a clear counterexample that refutes the claim made in the article (which is stated without evidence). The article should be revised in light of this. 174.2.168.156 (talk) 01:09, 18 October 2013 (UTC)

The definition used in that PDF is non-standard (though equivalent to the standard one in the case of finite groups). With the standard definition (as used in the article), the normalizer is always a subgroup. --Zundark (talk) 14:58, 18 October 2013 (UTC)

(edit conflict) Hi: the proof given at the link is not a disproof because it uses a nonstandard definition of normalizer. The definition used there is $a\in N_{G}(H)\iff aHa^{-1}\subseteq H$, whereas it should actually be (as it is in the article and in most texts) $a\in N_{G}(H)\iff aHa^{-1}=H$. As you can see, for the given a and H in that paper, $aHa^{-1}\subsetneq H$, so it is not in the (standard) normalizer. Rschwieb (talk) 15:21, 18 October 2013 (UTC)

Commutant

This is the same thing despite that article's claim to the contrary. There are plenty of sources defining centralizer for semigroups [2][3]. JMP EAX (talk) 07:08, 24 August 2014 (UTC)

Merge  Done given the lack of opposition after a week. JMP EAX (talk) 10:01, 3 September 2014 (UTC)

semigroups

This has a lot of interesting material. JMP EAX (talk) 09:26, 24 August 2014 (UTC)

And so does this. JMP EAX (talk) 09:30, 24 August 2014 (UTC)