Conjugate Elements — Definition, Formula & Examples

Conjugate elements are two elements in a group that are related by an inner transformation — one can be turned into the other by 'sandwiching' it with a third element and its inverse. This is a group theory concept, distinct from complex conjugates in algebra.

In a group $G$ , elements $a$ and $b$ are said to be conjugate if there exists an element $g \in G$ such that $b = g a g^{-1}$ . Conjugacy is an equivalence relation on $G$ , partitioning it into disjoint conjugacy classes.

Key Formula

b = g \, a \, g^{-1}

Where:

$a$ = An element of the group G
$b$ = An element conjugate to a
$g$ = An element of G that conjugates a to b
$g^{-1}$ = The inverse of g in the group G

How It Works

To check whether two elements

a

and

b

are conjugate in a group

G

, you search for some

g \in G

satisfying

b = gag^{-1}

. If such a

g

exists,

a

and

b

share important structural properties: they have the same order, and in symmetric groups they have the same cycle type. The set of all elements conjugate to a given element

a

is called the conjugacy class of

a

. In an abelian group, every conjugacy class contains exactly one element, since

gag^{-1} = a

for all

g

Worked Example

Problem: In the symmetric group

S_3

, determine whether the permutations

a = (1\;2)

and

b = (1\;3)

are conjugate.

Step 1: Try

g = (2\;3)

and compute

g \, a \, g^{-1}

. Since

(2\;3)

is its own inverse,

g^{-1} = (2\;3)

g \, a \, g^{-1} = (2\;3)(1\;2)(2\;3)

Step 2: Evaluate by applying permutations right to left. Trace each element:

1 \mapsto 1 \mapsto 2 \mapsto 3

and

3 \mapsto 2 \mapsto 1 \mapsto 1

, while

2 \mapsto 3 \mapsto 3 \mapsto 2

(2\;3)(1\;2)(2\;3) = (1\;3)

Step 3: The result equals

b

, confirming the conjugacy relation.

b = g \, a \, g^{-1} \quad \checkmark

Answer: Yes,

(1\;2)

and

(1\;3)

are conjugate in

S_3

via

g = (2\;3)

Why It Matters

Conjugacy classes are central to the structure theory of finite groups and appear directly in the class equation, which is used to prove results like the existence of nontrivial centers in p-groups. In representation theory, the number of irreducible representations of a finite group equals the number of its conjugacy classes, making this concept essential in abstract algebra courses.

Common Mistakes

Mistake: Confusing conjugate elements in group theory with complex conjugates (e.g.,

a + bi

and

a - bi

Correction: These are entirely different concepts. Conjugate elements require a group operation and a conjugating element

g

; complex conjugates involve negating the imaginary part of a complex number.