Definition:Conjugate (Group Theory)/Element/Also defined as

Definition

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

Some sources define the conjugate of $x$ by $a$ in $G$ as:

$x \sim y \iff \exists a \in G: x \circ a = a \circ y$

or:

$x \sim y \iff \exists a \in G: a^{-1} \circ x \circ a = y$

This is clearly equivalent to the other definition by noting that if $a \in G$ then $a^{-1} \in G$ also.

Also known as

Some sources refer to the conjugate of $x$ as the transform of $x$.

Some sources refer to conjugacy as conjugation.

Also see

• Equivalence of Definitions of Conjugate of Group Element

• Results about conjugacy can be found here.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.5$. Groups acting on sets: Example $103$
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.5$. Orbits: Example $119$
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Exercise $5.16$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions: Examples of group actions: $\text{(v)}$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conjugate elements