# Definition:Conjugate (Group Theory)/Subset

< Definition:Conjugate (Group Theory)(Redirected from Definition:Conjugate of Group Subset)

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## Contents

## Definition

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.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 6.6$. Normal subgroups: Example $124$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 35 \gamma$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 45$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Definition $7.1$