Group Example: x inv c y

Theorem
Let $\struct {G, \circ}$ be a group.

Let $c \in G$.

We define a new product $*$ on $G$ as:


 * $\forall x, y \in G: x * y = x \circ c^{-1} \circ y$

Then $\struct {G, *}$ is a group.

$G \, 0$: Closure
Let $x, y \in G$.

Then:
 * $\forall x * y = x \circ c^{-1} \circ y \in G$ as $c^{-1} \in G$

thus demonstrating that $\struct {G, *}$ is closed.

$G \, 1$: Associativity
Let $x, y, z \in G$.

thus demonstrating that $\struct {G, *}$ is associative.

$G \, 2$: Identity
Let $x \in G$.

So $c$ serves as the identity in $\struct {G, *}$.

$G \, 3$: Inverses
Let $x \in G$.

We need to find $y \in G$ such that $x * y = c \implies x \circ c^{-1} \circ y = c$.

Thus the inverse of $x$ under the operation $*$ is $c \circ x^{-1} \circ c$ where $x^{-1}$ is the inverse of $x$ under $\circ$.

All criteria have been demonstrated to be fulfilled, and so $\struct {G, *}$ is a group.

Also see

 * Definition:Heap