Inner automorphism

In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup gives rise to the concept of the outer automorphism group. In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the conjugating element. These inner automorphisms form a subgroup of the automorphism group, and the quotient of the automorphism group by this subgroup gives rise to the concept of the outer automorphism group. If G is a group (or a ring) and g is an element of G (if G is a ring, then g must be a unit), then the function φ g : G ⟶ G φ g ( x ) := g − 1 x g {displaystyle {egin{aligned}varphi _{g}colon G&longrightarrow G\varphi _{g}(x)&:=g^{-1}xgend{aligned}}} is called (right) conjugation by g (see also conjugacy class). This function is a homomorphism of G: for all x 1 , x 2 ∈ G {displaystyle x_{1},x_{2}in G} , φ g ( x 1 x 2 ) = g − 1 x 1 x 2 g = ( g − 1 x 1 g ) ( g − 1 x 2 g ) = φ g ( x 1 ) φ g ( x 2 ) {displaystyle varphi _{g}(x_{1}x_{2})=g^{-1}x_{1}x_{2}g=(g^{-1}x_{1}g)(g^{-1}x_{2}g)=varphi _{g}(x_{1})varphi _{g}(x_{2})} where the second equality is given by the insertion of the identity between x 1 {displaystyle x_{1}} and x 2 {displaystyle x_{2}} . Furthermore, it has a left and right inverse, namely φ g − 1 {displaystyle varphi _{g^{-1}}} . Thus, φ g {displaystyle varphi _{g}} is bijective, and so an isomorphism of G onto itself, i.e. an automorphism. An inner automorphism is any automorphism that arises from conjugation. When discussing right conjugation, the expression g − 1 x g {displaystyle g^{-1}xg} is often denoted exponentially by x g {displaystyle x^{g}} . This notation is used because composition of conjugations is associative: ( x g 1 ) g 2 = x g 1 g 2 {displaystyle (x^{g_{1}})^{g_{2}}=x^{g_{1}g_{2}}} for all g 1 , g 2 ∈ G {displaystyle g_{1},g_{2}in G} . This shows that conjugation gives a right action of G on itself. The composition of two inner automorphisms is again an inner automorphism, and with this operation, the collection of all inner automorphisms of G is a group, the inner automorphism group of G denoted Inn(G). Inn(G) is a normal subgroup of the full automorphism group Aut(G) of G. The outer automorphism group, Out(G) is the quotient group