Talk:Grushko theorem

Latest comment: 16 years ago by JackSchmidt in topic permutation of conjugacy classes

Decomposition theorem

edit

A nice piece. It may be useful to include a mention of Grushko's decomposition theorem (on the uniqueness properties of decomposition as a free product). Katzmik (talk) 10:14, 5 August 2008 (UTC)Reply

Yes, thank you, good point! Will do. Nsk92 (talk) 14:34, 5 August 2008 (UTC)Reply
OK, done. Thanks again for the suggestion.Nsk92 (talk) 15:43, 5 August 2008 (UTC)Reply

permutation of conjugacy classes

edit

The current statement of the Grushko decomposition theorem is a but confusing. "Unique up to permitation of their conjugacy classes" may be shorter than the previous version but is less clear. Katzmik (talk) 11:14, 6 August 2008 (UTC)Reply

That is how the uniqueness statement is usually expressed in the literature. What it means more precisely is the following: if   is another such decomposition then k = r, s = t, and there exists a permutation   such that for each   the subgroups   and   are conjugate in G. I thought that saying this explicitly is too fussy and takes too much space, but maybe you are right and one should include this anyway for clarification... Nsk92 (talk) 11:40, 6 August 2008 (UTC)Reply
We used the decomposition theorem in a recent paper. From my experience, having a precise statement can be extremelly helpful. I added your sentence to the article. I feel that shortening it creates fatal ambiguities. Katzmik (talk) 12:12, 6 August 2008 (UTC)Reply
OK, sounds good to me. Nsk92 (talk) 13:10, 6 August 2008 (UTC)Reply
Having a precise "up to" statement is usually good. Wikipedia only needs to be an encyclopedia, not a reference textbook or handbook. Ideally, we just find an amazingly clear statement in the literature, and copy it with citation. The "copying it" part is prone to error, but the citation part should always be useful. JackSchmidt (talk) 14:46, 6 August 2008 (UTC)Reply