I am trying to perform some SEO on my blog.
I want to to keep sharing the blog on FaceBook and other social networks, sharing the link on different groups (i.e., groups related to my blog).
Is it fine or does this constitute spamming?
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My advice would be to put "share this"-style links on each post, tweet each post as you do it and on Facebook there is a feature where you can submit an RSS feed and when it is updated it will post as a note (it might not be a note, but I'm pretty sure it is).
Other than that, get your link into any signatures on forums (I don't think Stack Overflow allows signatures under each post, but I notice that you haven't added your website to your profile on here - anything helps)
It's fine! Matt Cutts (the Google alter ego) himself suggests to spread the word around about your site/blog on social networks.
Obviously there is a limit between just sharing it around and pushing it everywhere, but IMHO you wouldn't have to worry even in pushing it a bit cause I don't think Google is gonna pick on you unless you really do a huge sapmmy-push like creating thousands (1) of false accounts on Twitter/FB just to push it around. But creating those amount of accounts would take you maybe the rest of your life, so you better ask yourself if it would be worth it. Moreover FB/Twitter might pick up on you doing spam on their respective sites before Google. :)
(1) thousands it's just my opinion, I have no idea of how much you have to push it around before Google picks on you as a spammer, but probably no one ever done a test because it would take tto musch.
You are right, you do not want to spam people. However spreading the word on social sites/apps will help your site's SEO.
Some other things you can do to boost your ranking are:
The most important part is to continually providing value. Over time people will start linking to your site which really helps your ranking.
Never ever submit your site address to an app, site or whatever that promises to mass submit your site everywhere! This could really hurt your rankings.
Hope that helps and good luck!