# What's that process called where you form new group law by $x \star y = x \cdot a \cdot y$ for some $a$ in the group?

Let $A$ be an abelian group. We can form new groups $(A, \cdot a \cdot)$ where $a$ is any element of $A$. Choosing $a = 1$ the identity of $A$ gives $A$ itself.

Clearly, associativity comes from associativity and commutativity of $A$.

Identity is $x \cdot a \cdot e = x$ or $e = a^{-1}$.

Inverse is $x \cdot a \cdot y = a^{-1}$ or $y = a^{-1} x^{-1} a^{-1}$.

What is this group formation process called and do the groups relate back to $A$?

I've seen it somewhere, and can't find where again.

• $A$ doesn't have to be Abelian. This whole game has something to do with torsors. – Lord Shark the Unknown Apr 11 '18 at 5:21
• There is an isomorphism $(A,\cdot)\to(A, \cdot a\cdot)$ given by $x\mapsto xa^{-1}$. For groups like vector spaces, this is known as "moving the origin to $a$". – Arthur Apr 11 '18 at 5:58
• Also known as transport of structure. – Andreas Caranti Apr 11 '18 at 14:21
• For a recent paper see here, with $X\star Y:=XAY$, where it is called "sandwich". – Dietrich Burde Apr 11 '18 at 18:57

For $K$-algebras $(A,\cdot)$ with multiplication $x\cdot y$ a new multiplication $x\circ_z y$ depending on a fixed $z$ is called a homotope, or a mutation.