As part of a larger problem I am working on, I need to come up with an example as follows: A set of exactly 4 elements with a binary operation. The set plus the operation have to comply with all the requirements of a group with the exception of the last one. I.e. Invertibility. It can be a known operation or a made up one * with a table. It is important though that invertibilty will not exist for a least one of the elements. In other words the 4 elements set, exhibits the first 3 qualities of a group but not the last one.
I tried thinking of a known operation such as addition but could not find an example that will maintain closure. With a made up operation and a table it is very difficult to check for associativity so I'm lost there as well.
Any help would be appreciated. Cheers