I tried lot of plugins, which intend to connect your wordpress site as close as possible with Facebook. After successes and failures, I think I found two best ones.
Simple Facebook Connect – very simple and neat plugin, which allow to login, comment, share and post links to Facebook. The only issue I found – it is not compatible with e-Commerce plugin. It was pretty critical for one of my client.
Further searches brought me to another plugin Add Link to Facebook. Exactly what I was looking for. It allows to post to any fan page and group you are allowed to post. For the feature to post to fan pages and groups you need to donate at least 25 USD, but I consider it worth fully. And finally it is compatible with e-Commerce without any problems.