Group Example: x inv c y
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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
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Exercise $3$