conjugation

introduction

%%visits: 3 conjugation makes way for normal_subgroups, which make the left coset equal to the right coset

Think, similar elements. ## intuition relabel keeps the cycle the same

conjugation :~ think of changing_the_basis of_a_matrix or similar_matrices, it is the same formula but applied to the groups. We do this for the symmetric group.

conjugation relates two elements if they are symmetric in a certain way. Ones that have similar structure (like order). They help with being able to sort elements out and deal with them together, which is useful when there are a large number of elements.

Conjugation is and equivalence relation (we use group properties) - Reflexive: a a as e ∈ G hence eae⁻¹ = a - Symmetric: a b ⇔ b a as a b ⟹ ∃g ∈ G : a = gbg⁻¹ ⟹ g⁻¹ag = b ⟹ b a - Transitive: a b, b c ⟹ a c as a = gbg⁻¹, b = g^′cg^′−1 ⟹ a = gg^′cg^′−1g⁻¹ = gg^′c(gg^′)⁻¹ due to closure, a c

conjugation changes the labels of cycles. Consider 3-cycles. a = (132) and b = (123) may look different, but they have the same underlying structure, if we just relabelled the element, they are similar. Conjugation, loosely changing_the_basis. In other words, swap 2 ↔︎ 3. so (23)(123)(23) = (132)

rigour

conjugatio := Given a group G, we say that a ∈ G is conjugate to b ∈ G if there exists g ∈ G : agbg⁻¹

Let a, b ∈ G conjugation by a of b is : aba⁻¹

exam clinic

examples and non-examples

resources

tags :math:

backlinks

• index_of_files.pd