Group Example: x inv c y

From ProofWiki
Jump to navigation Jump to search

Theorem

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

Let $c \in G$.

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

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


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


Proof

Group Axiom $\text G 0$: 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.

$\Box$


Group Axiom $\text G 1$: Associativity

Let $x, y, z \in G$.

\(\ds x * \paren {y * z}\) \(=\) \(\ds x \circ c^{-1} \circ \paren {y \circ c^{-1} \circ z}\) Definition of $*$
\(\ds \) \(=\) \(\ds \paren {x \circ c^{-1} \circ y} \circ c^{-1} \circ z\) Associativity of $\circ$
\(\ds \) \(=\) \(\ds \paren {x * y} * z\) Definition of $*$

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

$\Box$


Group Axiom $\text G 2$: Existence of Identity Element

Let $x \in G$.

\(\ds x * c\) \(=\) \(\ds x \circ c^{-1} \circ c = x\)
\(\ds c * x\) \(=\) \(\ds c \circ c^{-1} \circ x = x\)

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

$\Box$


Group Axiom $\text G 3$: Existence of Inverse Element

Let $x \in G$.

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

\(\ds x * y\) \(=\) \(\ds c\)
\(\ds \leadsto \ \ \) \(\ds x \circ c^{-1} \circ y\) \(=\) \(\ds c\)
\(\ds \leadsto \ \ \) \(\ds c^{-1} \circ y\) \(=\) \(\ds x^{-1} \circ c\)
\(\ds \leadsto \ \ \) \(\ds y\) \(=\) \(\ds c \circ x^{-1} \circ c\)


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

$\Box$


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

$\blacksquare$


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Group_Example:_x_inv_c_y&oldid=623826"