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For example, the page "A" at first position in the SERP, if i use the 301 redirect from page "A" to "A1", the page "A1" retain at first position or not.
If you use a test environment and take care that everything is technically correct before you make the change live, you have the least possibility of major problems. Make sure that you have no "chains" of redirects - the best practice is just one "jump" and you've arrived.
So if you already have canonical fixes in place, for example, makes sure that they work like this:
[no-www long url] >> [with-www short url]
[no-www-long url] >> [with-www long url] >> [with-www short url]
We have a disucssion on this topic in the Hot Topics area [webmasterworld.com], which is always pinned to the top of this forum's index page. It's a bit old, but it still applies.
Has anyone actually seen that their SERP actually changes when they perform such a change?
The reason I am hesitating is that we have 1M+ pages indexed and trying to change all those, and don't make a mistake is not trivial.
However, I have NOT heard if anyone has an improvement in SERP after they moved they redirected the "bad URL" to the "good URL" structures. The best people are saying they got what they had before, pretty much. So if nobody is seeing huge improvements, then why risk the change?
The shorter url does seem to draw clicks more easily, even with the same rankings. But all my comments are assuming technical precision in all areas. Technical errors in a rewrite or redirect scheme can mess with your rankings quite severly.