Talk:Core (group theory)

Notation

It would be nice to include a symbolic version of the definition (i.e. with an intersection symbol) but I'm not sure how to do this!

Your wish is my command. I don't know if the notation is standard; I just made it up since I didn't have a reference on hand. - Gauge 00:34, 19 August 2006 (UTC)

Wikify

Careful with just reverting an edit. There were significant additions to the article deleted by that revert (basically a citation for your {{Fact}}, some wikilinks, some grammar). I have merged in your grammar corrections and most of the improvements. There are still some changes I have backed out, but hopefully they are easier to agree on:

1. The reordering of paragraphs seems unwise as suddenly a concept is used before it is defined.
2. Giving every paragraph its own subsection also seems overkill. Perhaps when the article is longer a more detailed structure is useful, but for now this is just a stub -- indeed just two stubs with basically the same name.
3. Linking orbit would be excellent except that it is just a redirect to the already linked group action.
4. "In mathematics, especially field name" is probably not better than "In group theory--a branch of mathematics", BUT it is a standard idiom used on wikipedia so just for consistency I put it back.

I think those are the only "reverts", but adding sections and reordering paragraphs really messes up the wikipedia differ, so I apologize if I too missed some of your improvements.

To be clear, neither of my last two edits have been "undo", each has been an attempt to incorporate your concerns into the article while making other obvious improvements. JackSchmidt (talk) 16:14, 7 December 2007 (UTC)

Apologies for my errors/omissions. I've finished changing the article now, so please go ahead and amend it as you now see fit. reetep (talk) 17:16, 7 December 2007 (UTC)

I've repeated my edits by introducing them one by one in small increments. I'm sure you'll approve of each one, and hence the final result. You can see how similar the final result is to what I did in the first place. reetep (talk) —Preceding comment was added at 11:37, 10 December 2007 (UTC)

Thanks for separating out the edits. While overall I don't agree with them, it is much easier to accept per edit. I think my reasons for omitting the four changes above are reasonable, but the edits don't harm the article. In case you want to hear why I disagree:

1. the reordering of solvable radical will make it harder to explain how normal cores and p'-cores are the same thing since some of it will now only hold for odd cores -- in effect there are 3 basic definitions on this page, normal cores, pi-cores, and solvable-pi-cores, but the notation for the latter two are the same.
2. the amount of screen space devoted to section headings is almost equal to the amount of screen space for content, but perhaps the real solution is to include more material, not less structure
3. orbit is still a useless/duplicate link, but perhaps someone will improve the orbit redirect
4. the lead no longer sets up the expectation that normal cores are defined for subgroups of a group and have no meaning merely in the context of a group (and it doesn't match the leads it links to, but perhaps that is a silly concern).

There was also a grammar fix near the modular representation theory link. The original text was poorly worded, but your fix removed the parallelism. I adjusted the wording to make it clear that normal cores are kernels of set actions and p-cores and p'-cores are kernels for vector space actions. JackSchmidt (talk) 14:24, 10 December 2007 (UTC)

Sources

The current text is too dense for citations, so I thought I'd remind myself and/or others of some of the sources:

• Huppert:
  • Kap. IV: p-nilpotent groups
  • Kap. VI: solvable stuff,
    • Definition of pi-stuff, pi-separable, pi-soluble
    • Satz 5.4: intersections of centralizers of (p-)chief factors are the (p-)nilpotent radical.
    • §6: p-length and upper p-series, more on p-nilpotent
• Huppert–Blackburn FG2:
  • Terminology: p-nilpotent, p-core, p′-core, upper p-series, etc.
  • Kap VII:
    • Theorem 13.4: p-core is intersection of kernels in char p
    • Theorem 13.7: p-constrained: principal block iff p'-core acts trivially
    • Theorem 14.6: p′-core is intersection for a block
    • Theorem 14.8: p',p-core is intersection of kernels in char p for the principal block —Preceding unsigned comment added by Parksj (talk • contribs) 12:36, 11 June 2010 (UTC)
    • Theorem 14.9: p-nilpotent iff IBr(B0)=1 iff IBr(B)=1 for all blocks B
    • Theorem 15.5: complemented p-chief factors are in second head of proj cover of trivial rep
    • Theorem 15.8: p-chief factors are comp factors are comp factors of proj cover of trivial rep
• Huppert–Blackburn FG3:
  • Kap X
    • § 13: generalized fitting, quasinilpotent
    • § 14: generalized p′-cores
• Navarro:
  • 6.10: p'-core is intersection of kernels in block (stated clearly)
  • 6.12: p-core is intersection of modular kernels in block
  • 6.13: p-nilpotent iff IBr(B0)=1

Doerk–Hawkes, Chapter A and B have marvelously concise summaries of the three Hupperts. TODO: p*-core, and quasi-p-nilpotent stuff. Give p-nilpotent and p-soluble groups their own article. lower p-series. JackSchmidt (talk) 04:19, 6 March 2009 (UTC)