Two elements \(x, y\) of a group \(G\) are conjugate if there is some \(h \in G\) such that \(hxh^{-1} = y\).

Conjugacy as “changing the worldview”

Conjugating by \(h\) is equivalent to “viewing the world through \(h\)’s eyes”. This is most easily demonstrated in the symmetric group, where it is a fact that if \($\sigma = (a_{11} a_{12} \dots a_{1 n_1})(a_{21} \dots a_{2 n_2}) \dots (a_{k 1} a_{k 2} \dots a_{k n_k})\)$ and \(\tau \in S_n\), then \($\tau \sigma \tau^{-1} = (\tau(a_{11}) \tau(a_{12}) \dots \tau(a_{1 n_1}))(\tau(a_{21}) \dots \tau(a_{2 n_2})) \dots (\tau(a_{k 1}) \tau(a_{k 2}) \dots \tau(a_{k n_k}))\)$

That is, conjugating by \(\tau\) has “caused us to view \(\sigma\) from the point of view of \(\tau\)”.

Similarly, in the dihedral group \(D_{2n}\) on \(n\) vertices, conjugation of the rotation by a reflection yields the inverse of the rotation: it is “the rotation, but viewed as acting on the reflected polygon”. Equivalently, if the polygon is sitting on a glass table, conjugating the rotation by a reflection makes the rotation act “as if we had moved our head under the table to look upwards first”.

In general, if \(G\) is a group which acts as (some of) the symmetries of a certain object \(X\) noteWhich we can always view as being the case. then conjugation of \(g \in G\) by \(h \in G\) produces a symmetry \(hgh^{-1}\) which acts in the same way as \(g\) does, but on a copy of \(X\) which has already been permuted by \(h\).

Closure under conjugation

If a subgroup \(H\) of \(G\) is closed under conjugation by elements of \(G\), then \(H\) is a normal subgroup. The concept of a normal subgroup is extremely important in group theory.

Conjugation action

Conjugation forms a action. Formally, let \(G\) act on itself: \(\rho: G \times G \to G\), with \(\rho(g, k) = g k g^{-1}\). It is an exercise to show that this is indeed an action. %%hidden(Show solution): We need to show that the identity acts trivially, and that products may be broken up to act individually.

• \(\rho(gh, k) = (gh)k(gh)^{-1} = ghkh^{-1}g^{-1} = g \rho(h, k) g^{-1} = \rho(g, \rho(h, k))\);

• \(\rho(e, k) = eke^{-1} = k\). %%

The stabiliser of this action, \(\mathrm{Stab}_G(g)\) for some fixed \(g \in G\), is the set of all elements such that \(kgk^{-1} = g\): that is, such that \(kg = gk\). Equivalently, it is the centraliser of \(g\) in \(G\): it is the subgroup of all elements which commute with \(G\).

The orbit of the action, \(\mathrm{Orb}_G(g)\) for some fixed \(g \in G\), is the conjugacy class of \(g\) in \(G\). By the orbit-stabiliser theorem, this immediately gives that the size of a conjugacy class divides the order of the parent group. <div>