Definition:Conjugate (Group Theory)/Subset

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Definition

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

Let $S \subseteq G, a \in G$.

Then the $G$-conjugate of $S$ by $a$ is:

$S^a := \left\{{y \in G: \exists x \in S: y = a \circ x \circ a^{-1}}\right\} = a \circ S \circ a^{-1}$

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 (as is usual), we usually just refer to the conjugate of $S$ by $a$.

Similarly to the alternative definition for group elements, the concept of set conjugacy can be defined as:

$S^a := \left\{{y \in G: \exists x \in S: y = a^{-1} \circ x \circ a}\right\} = a^{-1} \circ S \circ a$

There is a subtle difference between the definitions.