# Group in which all elements are conjugate [duplicate]

Two elements $$a$$ and $$b$$ of a group $$G$$ are conjugate if there is an element $$g$$ in the group such that $$b = g^{–1}ag$$.

What can we say about a group $$G$$ in which all its elements are conjugate? Does it have any special properties? Can there exist such a non-trivial group $$G$$?

• To make it more interesting, consider replacing "all elements are conjugate" with "all non-identity elements are conjugate". Then, the cyclic group of order $2$ would be such a group. May 3 '20 at 23:39
• @saltandpepper: If the group is abelian, then every element is conjugate only to itself. May 3 '20 at 23:47
• A group in which every element is conjugate is trivial, since $e$ can only be conjugate to itself. A group with exactly two conjugacy classes, one for the identity and one for every other element, is either cyclic of order $2$, or must be infinite, and the constructions are difficult, and the question is then a duplicate. May 3 '20 at 23:47

That can't happen unless the group is trivial. The class equation always consists of at least one $$1$$, corresponding to the class of the identity.