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 $$G$$ such that:

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

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

G0: Closure
Let $$x, y \in G$$.

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

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

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

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

And thus we see that $$\left({G, *}\right)$$ is a group.