I am looking to find examples of the following if they exist or not. A group with exactly 7 elements and a binary operation on a set $S$ with identity, such that not every element $a \in S$ has an inverse. Its a question that I am really stuck on.
2026-03-25 23:24:49.1774481089
Can someone please help me find examples of the following if they exist or not.
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Take $S=\{0,1,2,3,4,5,6\}$ and define multiplication $\odot$ by
$a\odot b := a\cdot b \text{ mod } 7$
Clearly $1$ is the identity with respect to that multiplication. But $0$ has no inverse.