Define a binary composition ( operation ) $\star$ such that $\langle G,\star\rangle$ is a group with $a$ as its identity.

Question

Let $G$ be a group and $a\in G $ be any non identity element i.e. $a\neq e$. Define a binary composition ( operation ) $\star$ such that $\langle G,\star \rangle$ is a group with $a$ as its identity.

I am new to the class of the Abstract Algebra and this was a question which was asked to solved on the third day. I don't think I understood the concept and question well enough to solve the problem.

Kindly help with the concepts and provide good explanation to the steps to reduce further confusion. Moreover, suggestions regarding a good book to read will be also helpful.

Thanks in advance!

Edit: What I think is that the question missed a very crucial information that, it should be G be a group under _________ operation. Only then it will be possible to solve. Correct me if I am wrong !!

Answer 1

Hint: Define $g*h = ga^{-1}h$, where the product on the right uses the original group structure.