# Relation between a set being closed under a binary operation and the set being a group under that binary operation

If a set $S$ is not closed under some binary operation $\star$, is it true that $S$ cannot be a group under $\star$?

• Did you figure out your dihedral group question? I was writing an answer to it. – anon Oct 9 '13 at 14:36
• Sorry, yeah I figured it out after staring at $D_8$'s lattice structure for a bit. – Alex Oct 9 '13 at 14:38
• If you're looking at the lattice structure of $D_8$ hosted on a website in order to figure out $D_{2n}/\langle r\rangle$, it may be problematic; it's necessary to be able to figure out what quotient groups are and what their elements look like on your own, especially for basic groups like dihedral ones. Here is what I was going to write. – anon Oct 9 '13 at 15:12