if , then is defined iff ,
if and , then ,
if , then ,
if and such that , then for all .
If , then is a group.
Now, by condition 3, given , there is such that , so that for all by condition 4. In other words, is a left identity of . Again, by condition 3, for every , there is a such that . So , so, by condition 4, for all . In particular, set , we get . Hence, is a two-sided identity, and is a monoid.
Finally, by condition 3, for every , there are , such that . So, , showing that has a two-sided inverse. This means that is a group. ∎
For a non-trivial example of a mixed group, let be a group and a subgroup of . Define a new multiplication on as follows: is defined iff , and if is defined, it is defined as , the group multiplication of and . Then is a mixed group. Clearly, associativity of is automatically satisfied. Next, pick any , then, for any , and are both elements of , so that , and condition 3 is also satisfied. Finally, if and such that , then is the multiplicative identity of , clearly for all .
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