# Group conjugate

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 $$X$$ note 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.

knows-requisite(Group action):

# 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>

Parents:

• Group

The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.