Four Objects Groups: Is x^4=e Sufficient?

[yetar]

I know that there are only two four objects groups.
However, I want a term that will be true if and only M is a four objects group.
Will saying that for every x in M, x^4=e will be enought? (Apart from the fact the also a 2 objects group is such a group)
Or is it possible that for every x in M, x^4=e also for groups with more then 4 objects in it?

Thanks in advance.

 

[matt grime]

There are infinitely many groups that satisfy x^4=4 for all x in G: any product of 4-groups for instance.

 

[tacman]

Yes, saying that for every x in M, x^4=e is sufficient to determine if M is a four objects group. This is because the operation of exponentiation is closed in a group, meaning that if x is an element in the group, x^4 will also be an element in the group. Therefore, if every element in M satisfies the condition x^4=e, then M must have exactly four elements. This is true regardless of the size of the group, as long as it is a four objects group. So, this statement can be used to determine if a group is a four objects group or not.