Group Example: x inv c y

Theorem
Let $\left({G, \circ}\right)$ be a group, and 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 $\left({G, *}\right)$ is a group.

G0: 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 $\left({G, *}\right)$ is closed.

G1: Associativity
Let $x, y, z \in G$.

thus demonstrating that $\left({G, *}\right)$ is associative.

G2: Identity
Let $x \in G$.


 * $x * c = x \circ c^{-1} \circ c = x$
 * $c * x = c \circ c^{-1} \circ x = x$

So $c$ serves as the identity.

G3: 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 $\left({G, *}\right)$ is a group.