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

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 !!

• @Amir I didn't get your hint. Could you be a little more specific, why $\star : {G} \times {G}\mapsto {G}$ ?? – shakunimama Aug 8 '16 at 16:00

## 1 Answer

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

• How it is that that on the right uses the original group structure. Could you be a little more specific, Please !! I am having problem understanding. – shakunimama Aug 8 '16 at 15:59
• That is not a hint, but the solution. I think it would be better to say how one gets to this definition. @1729, say $(G, \cdot)$ is a group, then you have nothing more then the group operation $\cdot : G \times G \to G$, and multiplication with some group element $\cdot g : G \to G$ to define your new operation. Then note that $e \star a = e$ and you get to Alex' definition. – mcd Aug 8 '16 at 16:00
• @MoritzDoll It's not a complete solution in the sense that one has to check that this does define a group structure, and that it satisfies the required property. – Alex Provost Aug 8 '16 at 16:03
• @MoritzDoll: In the question when it says ' Let G be a group and 'a' belongs to 'G' be any non identity element ' Does it mean that G is a group under multiplication, where a is not the identity element. If so, how to know about that ?? – shakunimama Aug 8 '16 at 16:04
• @1729 The group operation is usually referred to as "multiplication" by default. – Alex Provost Aug 8 '16 at 16:26