Talk:Grushko theorem

Decomposition theorem

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

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

G=B_{1}\ast \dots \ast B_{k}\ast F_{t}

is another such decomposition then k = r, s = t, and there exists a permutation

\sigma \in S_{r}

such that for each

i=1,\dots ,r

the subgroups

A_{i}

and

B_{\sigma (i)}

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]