Definition:Conjugate (Group Theory)/Subset

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

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 $a$.

Also defined as

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

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

There is a subtle difference between the definitions.

See, for example, Conjugate of Set by Group Product.

Some sources insist that this definition applies only to a subgroup of a group, not a general subset.

Also see

  • Results about conjugacy can be found here.