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If they did increase a ranking for those sites, that would certainly be a discriminating factor in ranking pages. Pages would not be ranked on content, but on the kiss-a$$ methods they use to link to google, etc.
A link to Google can't hurt you mind you. I link to them off the links page, also from several other pages, using the cobranding site search, which is kind of cool - Google results with your logo at the top:)
meaning whichever page the link to Google is on ... will lose out.
Not quite - pages that are also linked to *from* that page will lose out depending on how many other links are outbound from that page.
It simply dilutes the amount of page rank that is transferred via outbound links on that page by one extra link.
In answer to the original question, no it won't help you. Unless you're trying to rank for the keyword "google" in which case, good luck.
TJ