Group home is a subset of institution. Institutions aren't defined by size, they're defined by a particular power structure. You can have an institution with only one person if you do it right. (Or rather, wrong.) Calling larger institutions "true" institutions obscures the fact that the differences between different types (in terms of size and shape) are only cosmetic.
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"In my world it's a place of patterns and feel. In my world it's a haven for what is real. It's my world, nobody can steal it, but people like me, we live in the shadows." -Donna Williams