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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.

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    $\begingroup$ $A$ doesn't have to be Abelian. This whole game has something to do with torsors. $\endgroup$ – Lord Shark the Unknown Apr 11 '18 at 5:21
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    $\begingroup$ 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 $a$". $\endgroup$ – Arthur Apr 11 '18 at 5:58
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    $\begingroup$ Also known as transport of structure. $\endgroup$ – Andreas Caranti Apr 11 '18 at 14:21
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    $\begingroup$ For a recent paper see here, with $X\star Y:=XAY$, where it is called "sandwich". $\endgroup$ – Dietrich Burde Apr 11 '18 at 18:57
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I have seen this also for groups, but it seems to be more interesting for Lie algebras, see here and here. For groups it seems to be called variant, see the comments at the above questions, or a sandwich, e.g., Semigroups under a sandwich operation, Proc. Edinburgh Math. Soc. (Ser. 2) 26 (1983), 371-382.

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.

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