Definition:Conjugate (Group Theory)
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Definition
Let $\left({G, \circ}\right)$ be a group.
Conjugate of an Element
Let $\struct {G, \circ}$ be a group.
Definition 1
The conjugacy relation $\sim$ is defined on $G$ as:
- $\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x = y \circ a$
Definition 2
The conjugacy relation $\sim$ is defined on $G$ as:
- $\forall \tuple {x, y} \in G \times G: x \sim y \iff \exists a \in G: a \circ x \circ a^{-1} = y$
Conjugate of a Set
Let $S \subseteq G, a \in G$.
Then the $G$-conjugate of $S$ by $a$ is:
- $S^a := \set {y \in G: \exists x \in S: y = a \circ x \circ a^{-1} } = a \circ S \circ a^{-1}$
That is, $S^a$ is the set of all elements of $G$ that are the conjugates of elements of $S$ by $a$.
When $G$ is the only group under consideration, we usually just refer to the conjugate of $S$ by $a$.