Definition:Conjugacy Class
This page is about conjugacy class in the context of group theory. For other uses, see class.
Definition
The equivalence classes into which the conjugacy relation divides its group into are called conjugacy classes.
The conjugacy class of an element $x \in G$ can be denoted $\conjclass x$.
Also denoted as
Some authors denote a conjugacy class with the notation $\map {\operatorname {cl} } x$, but this can be confused with the notation for closure in the context of topology, so its use is not recommended.
Variants on the $\mathrm C$ motif can be seen: $C_x$ or $\map C x$ are fairly common.
Examples
Conjugacy Classes of Symmetric Group on $3$ Letters
Let $S_3$ denote the Symmetric Group on 3 Letters, whose Cayley table is given as:
- $\begin{array}{c|cccccc}\circ & e & (123) & (132) & (23) & (13) & (12) \\ \hline e & e & (123) & (132) & (23) & (13) & (12) \\ (123) & (123) & (132) & e & (13) & (12) & (23) \\ (132) & (132) & e & (123) & (12) & (23) & (13) \\ (23) & (23) & (12) & (13) & e & (132) & (123) \\ (13) & (13) & (23) & (12) & (123) & e & (132) \\ (12) & (12) & (13) & (23) & (132) & (123) & e \\ \end{array}$
The conjugacy classes of $S_3$ are:
\(\ds \) | \(\) | \(\ds \set e\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \set {\tuple {123}, \tuple {132} }\) | ||||||||||||
\(\ds \) | \(\) | \(\ds \set {\tuple {12}, \tuple {13}, \tuple {23} }\) |
Also see
- Results about conjugacy classes can be found here.
Technical Note
The $\LaTeX$ code for \(\conjclass {x}\) is \conjclass {x}
.
When the subscript is a single character, it is usual to omit the braces:
\conjclass x
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 6.5$. Orbits: Example $119$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 51$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 48$ Conjugacy
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.15$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conjugacy class
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): conjugacy class